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Номер 133, страница 44 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
I. Уравнения, неравенства с двумя переменными и их системы. 4. Неравенства с двумя переменными - номер 133, страница 44.
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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1102 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107643 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Неравенства с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "40" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "18" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107643 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "4. Неравенства с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "40" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:3 [ 0 => App\Models\Element {#1166 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276488 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1167 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "text" => "<p><strong>133.</strong> Запишите неравенство, множество решений которого показано на рисунке 18.</p><p><strong>а)</strong></p><svg width="200" height="150" viewBox="0 0 200 150"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="150" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="105" y="130" font-size="12">1</text><!-- Graph y = |x| --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><path d="M 60 110 L 100 150 L 140 110" stroke="black" fill="none" stroke-width="1"></path><text x="145" y="115" font-size="12">$y = |x|$</text><!-- Shading (vertical lines above y=|x|) --><line x1="100" y1="150" x2="100" y2="60" stroke="gray" stroke-width="0.5"></line><line x1="110" y1="140" x2="110" y2="50" stroke="gray" stroke-width="0.5"></line><line x1="120" y1="130" x2="120" y2="40" stroke="gray" stroke-width="0.5"></line><line x1="130" y1="120" x2="130" y2="30" stroke="gray" stroke-width="0.5"></line><line x1="140" y1="110" x2="140" y2="20" stroke="gray" stroke-width="0.5"></line><line x1="90" y1="140" x2="90" y2="50" stroke="gray" stroke-width="0.5"></line><line x1="80" y1="130" x2="80" y2="40" stroke="gray" stroke-width="0.5"></line><line x1="70" y1="120" x2="70" y2="30" stroke="gray" stroke-width="0.5"></line><line x1="60" y1="110" x2="60" y2="20" stroke="gray" stroke-width="0.5"></line></svg><p>$y \ge |x|$</p><p><strong>б)</strong></p><svg width="200" height="150" viewBox="0 0 200 150"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="150" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="140" y="155" font-size="12">2</text><text x="80" y="155" font-size="12">-1</text><text x="60" y="155" font-size="12">-2</text><text x="105" y="130" font-size="12">1</text><text x="105" y="110" font-size="12">2</text><!-- Graph y = |x^2 - 2| --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><!-- Points: (-2,2)->(60,110), (-sqrt(2),0)->(71.8,150), (-1,1)->(80,130), (0,2)->(100,110) (1,1)->(120,130), (sqrt(2),0)->(128.2,150), (2,2)->(140,110) --><path d="M 60 110 Q 65 130 71.8 150 L 80 130 Q 90 120 100 110 Q 110 120 120 130 L 128.2 150 Q 135 130 140 110" stroke="black" fill="none" stroke-width="1"></path><text x="145" y="115" font-size="12">$y = |x^2 - 2|$</text><!-- Shading (vertical lines between x-axis and curve) --><defs><pattern id="hatch" patternUnits="userSpaceOnUse" width="4" height="4"><path d="M-1,1 l2,-2 M0,4 l4,-4 M3,5 l2,-2" stroke="gray" stroke-width="0.5"/></pattern></defs><!-- The shaded path. From (-2,0) to (2,0) along x-axis and then along the curve. --><path d="M 60 150 L 60 110 Q 65 130 71.8 150 L 80 130 Q 90 120 100 110 Q 110 120 120 130 L 128.2 150 Q 135 130 140 110 L 140 150 Z" fill="url(#hatch)" stroke="none"></path></svg><p>$0 \le y \le |x^2 - 2|$</p><p><strong>в)</strong></p><svg width="200" height="260" viewBox="0 0 200 260"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="250" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="105" y="130" font-size="12">1</text><!-- Graph y = -2/x --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><!-- Points: (-2,1)->(60,130), (-1,2)->(80,110), (-0.5,4)->(90,70) --><!-- (0.5,-4)->(110,230), (1,-2)->(120,190), (2,-1)->(140,170) --><path d="M 60 130 Q 70 100 90 70" stroke="black" fill="none" stroke-width="1"></path><!-- Left branch --><path d="M 110 230 Q 130 200 140 170" stroke="black" fill="none" stroke-width="1"></path><!-- Right branch --><text x="125" y="125" font-size="12">$y = -\frac{2}{x}$</text><!-- Shading (vertical lines between x-axis and curve) --><defs><pattern id="hatch2" patternUnits="userSpaceOnUse" width="4" height="4"><path d="M-1,1 l2,-2 M0,4 l4,-4 M3,5 l2,-2" stroke="gray" stroke-width="0.5"/></pattern></defs><!-- Shaded region for x < 0: from y=0 (150) to y=-2/x (curve) --><path d="M 90 150 L 90 70 Q 70 100 60 130 L 60 150 Z" fill="url(#hatch2)" stroke="none"></path><!-- Shaded region for x > 0: from y=-2/x (curve) to y=0 (150) --><path d="M 110 150 L 110 230 Q 130 200 140 170 L 140 150 Z" fill="url(#hatch2)" stroke="none"></path></svg><p>$-2 \le xy \le 0$</p><p>Рисунок 18</p>" "img" => array:1 [ 0 => array:5 [ "name" => "133-1.jpg" "alt" => null "width" => "1620" "height" => 680 "path" => "/media/algebra_09/soltan-u19/0-1/133-1.webp?ts=1752834198" ] ] ] #original: array:7 [ "id" => 1276488 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "text" => "<p><strong>133.</strong> Запишите неравенство, множество решений которого показано на рисунке 18.</p><p><strong>а)</strong></p><svg width="200" height="150" viewBox="0 0 200 150"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="150" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="105" y="130" font-size="12">1</text><!-- Graph y = |x| --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><path d="M 60 110 L 100 150 L 140 110" stroke="black" fill="none" stroke-width="1"></path><text x="145" y="115" font-size="12">$y = |x|$</text><!-- Shading (vertical lines above y=|x|) --><line x1="100" y1="150" x2="100" y2="60" stroke="gray" stroke-width="0.5"></line><line x1="110" y1="140" x2="110" y2="50" stroke="gray" stroke-width="0.5"></line><line x1="120" y1="130" x2="120" y2="40" stroke="gray" stroke-width="0.5"></line><line x1="130" y1="120" x2="130" y2="30" stroke="gray" stroke-width="0.5"></line><line x1="140" y1="110" x2="140" y2="20" stroke="gray" stroke-width="0.5"></line><line x1="90" y1="140" x2="90" y2="50" stroke="gray" stroke-width="0.5"></line><line x1="80" y1="130" x2="80" y2="40" stroke="gray" stroke-width="0.5"></line><line x1="70" y1="120" x2="70" y2="30" stroke="gray" stroke-width="0.5"></line><line x1="60" y1="110" x2="60" y2="20" stroke="gray" stroke-width="0.5"></line></svg><p>$y \ge |x|$</p><p><strong>б)</strong></p><svg width="200" height="150" viewBox="0 0 200 150"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="150" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="140" y="155" font-size="12">2</text><text x="80" y="155" font-size="12">-1</text><text x="60" y="155" font-size="12">-2</text><text x="105" y="130" font-size="12">1</text><text x="105" y="110" font-size="12">2</text><!-- Graph y = |x^2 - 2| --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><!-- Points: (-2,2)->(60,110), (-sqrt(2),0)->(71.8,150), (-1,1)->(80,130), (0,2)->(100,110) (1,1)->(120,130), (sqrt(2),0)->(128.2,150), (2,2)->(140,110) --><path d="M 60 110 Q 65 130 71.8 150 L 80 130 Q 90 120 100 110 Q 110 120 120 130 L 128.2 150 Q 135 130 140 110" stroke="black" fill="none" stroke-width="1"></path><text x="145" y="115" font-size="12">$y = |x^2 - 2|$</text><!-- Shading (vertical lines between x-axis and curve) --><defs><pattern id="hatch" patternUnits="userSpaceOnUse" width="4" height="4"><path d="M-1,1 l2,-2 M0,4 l4,-4 M3,5 l2,-2" stroke="gray" stroke-width="0.5"/></pattern></defs><!-- The shaded path. From (-2,0) to (2,0) along x-axis and then along the curve. --><path d="M 60 150 L 60 110 Q 65 130 71.8 150 L 80 130 Q 90 120 100 110 Q 110 120 120 130 L 128.2 150 Q 135 130 140 110 L 140 150 Z" fill="url(#hatch)" stroke="none"></path></svg><p>$0 \le y \le |x^2 - 2|$</p><p><strong>в)</strong></p><svg width="200" height="260" viewBox="0 0 200 260"><!-- Axes --><line x1="20" y1="150" x2="180" y2="150" stroke="black" stroke-width="1"></line><!-- X-axis --><line x1="100" y1="20" x2="100" y2="250" stroke="black" stroke-width="1"></line><!-- Y-axis --><!-- Axis arrows --><path d="M 180 150 L 175 147 L 175 153 Z" fill="black"></path><path d="M 100 20 L 97 25 L 103 25 Z" fill="black"></path><!-- Labels --><text x="185" y="155" font-size="12">x</text><text x="105" y="25" font-size="12">y</text><text x="105" y="160" font-size="12">O</text><text x="120" y="155" font-size="12">1</text><text x="105" y="130" font-size="12">1</text><!-- Graph y = -2/x --><!-- Scale: 1 unit = 20 pixels. Origin O(100, 150). (x,y) -> (100+x*20, 150-y*20) --><!-- Points: (-2,1)->(60,130), (-1,2)->(80,110), (-0.5,4)->(90,70) --><!-- (0.5,-4)->(110,230), (1,-2)->(120,190), (2,-1)->(140,170) --><path d="M 60 130 Q 70 100 90 70" stroke="black" fill="none" stroke-width="1"></path><!-- Left branch --><path d="M 110 230 Q 130 200 140 170" stroke="black" fill="none" stroke-width="1"></path><!-- Right branch --><text x="125" y="125" font-size="12">$y = -\frac{2}{x}$</text><!-- Shading (vertical lines between x-axis and curve) --><defs><pattern id="hatch2" patternUnits="userSpaceOnUse" width="4" height="4"><path d="M-1,1 l2,-2 M0,4 l4,-4 M3,5 l2,-2" stroke="gray" stroke-width="0.5"/></pattern></defs><!-- Shaded region for x < 0: from y=0 (150) to y=-2/x (curve) --><path d="M 90 150 L 90 70 Q 70 100 60 130 L 60 150 Z" fill="url(#hatch2)" stroke="none"></path><!-- Shaded region for x > 0: from y=-2/x (curve) to y=0 (150) --><path d="M 110 150 L 110 230 Q 130 200 140 170 L 140 150 Z" fill="url(#hatch2)" stroke="none"></path></svg><p>$-2 \le xy \le 0$</p><p>Рисунок 18</p>" "img" => array:1 [ 0 => array:5 [ "name" => "133-1.jpg" "alt" => null "width" => "1620" "height" => 680 "path" => "/media/algebra_09/soltan-u19/0-1/133-1.webp?ts=1752834198" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1173 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1277766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "133-1.jpg" "alt" => null "width" => "508" "height" => 570 "path" => "/media/algebra_09/soltan-u19/1-1/133-1.webp?ts=1752830866" ] ] ] #original: array:6 [ "id" => 1277766 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "133-1.jpg" "alt" => null "width" => "508" "height" => 570 "path" => "/media/algebra_09/soltan-u19/1-1/133-1.webp?ts=1752830866" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1181 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1553734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Edition {#1182 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "text" => "<p><strong>а)</strong> На рисунке показана заштрихованная область, которая находится выше графика функции $y = |x|$. Граница области, то есть сам график $y = |x|$, изображена пунктирной линией. Это означает, что точки, лежащие непосредственно на линии, не являются частью решения. Таким образом, для любой точки $(x, y)$ из заштрихованной области должно выполняться строгое неравенство, где значение $y$ больше соответствующего значения на границе. Это приводит к неравенству $y > |x|$.<br>Для проверки можно взять произвольную точку из заштрихованной области, например, точку $(0, 1)$. Подставив ее координаты в полученное неравенство, получаем $1 > |0|$, или $1 > 0$, что является верным утверждением.<br>Ответ: $y > |x|$.</p><p><strong>б)</strong> Заштрихованная область на рисунке ограничена сверху графиком функции $y = |x^2 - 2|$ и снизу осью абсцисс ($y=0$). Обе границы изображены сплошными линиями, что указывает на то, что точки на границах включаются в множество решений. Следовательно, неравенства будут нестрогими.<br>Условие, что область находится ниже или на графике $y = |x^2 - 2|$, записывается как $y \le |x^2 - 2|$.<br>Условие, что область находится выше или на оси абсцисс, записывается как $y \ge 0$.<br>Оба этих условия должны выполняться одновременно, что приводит к системе неравенств. Эту систему можно также записать в виде двойного неравенства.<br>Проверим на контрольной точке $(0, 1)$ из заштрихованной области:<br>1) $1 \le |0^2 - 2| \implies 1 \le 2$ (верно).<br>2) $1 \ge 0$ (верно).<br>Оба условия выполнены.<br>Ответ: $\begin{cases} y \le |x^2 - 2| \\ y \ge 0 \end{cases}$ (или $0 \le y \le |x^2 - 2|$).</p><p><strong>в)</strong> На рисунке изображена область, ограниченная гиперболой $y = -\frac{2}{x}$ и осями координат ($x=0$, $y=0$). Все границы сплошные, поэтому неравенства будут нестрогими. Область состоит из двух частей.<br>1. Во второй координатной четверти ($x < 0, y > 0$): область находится между осью $x$ и гиперболой, то есть $0 \le y \le -\frac{2}{x}$. Умножим правую часть неравенства ($y \le -\frac{2}{x}$) на $x$. Так как $x < 0$, знак неравенства меняется на противоположный: $xy \ge -2$. Из левой части ($0 \le y$) при $x < 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.<br>2. В четвертой координатной четверти ($x > 0, y < 0$): область находится между гиперболой и осью $x$, то есть $-\frac{2}{x} \le y \le 0$. Умножим левую часть неравенства ($-\frac{2}{x} \le y$) на $x$. Так как $x > 0$, знак неравенства не меняется: $-2 \le xy$. Из правой части ($y \le 0$) при $x > 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.<br>Так как для обеих частей заштрихованной области получилось одно и то же неравенство, оно и является искомым.<br>Ответ: $-2 \le xy \le 0$.</p>" ] #original: array:6 [ "id" => 1553734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191} "task" => array:2 [ "refs" => "1107831" "type" => "task" ] "text" => "<p><strong>а)</strong> На рисунке показана заштрихованная область, которая находится выше графика функции $y = |x|$. Граница области, то есть сам график $y = |x|$, изображена пунктирной линией. Это означает, что точки, лежащие непосредственно на линии, не являются частью решения. Таким образом, для любой точки $(x, y)$ из заштрихованной области должно выполняться строгое неравенство, где значение $y$ больше соответствующего значения на границе. Это приводит к неравенству $y > |x|$.<br>Для проверки можно взять произвольную точку из заштрихованной области, например, точку $(0, 1)$. Подставив ее координаты в полученное неравенство, получаем $1 > |0|$, или $1 > 0$, что является верным утверждением.<br>Ответ: $y > |x|$.</p><p><strong>б)</strong> Заштрихованная область на рисунке ограничена сверху графиком функции $y = |x^2 - 2|$ и снизу осью абсцисс ($y=0$). Обе границы изображены сплошными линиями, что указывает на то, что точки на границах включаются в множество решений. Следовательно, неравенства будут нестрогими.<br>Условие, что область находится ниже или на графике $y = |x^2 - 2|$, записывается как $y \le |x^2 - 2|$.<br>Условие, что область находится выше или на оси абсцисс, записывается как $y \ge 0$.<br>Оба этих условия должны выполняться одновременно, что приводит к системе неравенств. Эту систему можно также записать в виде двойного неравенства.<br>Проверим на контрольной точке $(0, 1)$ из заштрихованной области:<br>1) $1 \le |0^2 - 2| \implies 1 \le 2$ (верно).<br>2) $1 \ge 0$ (верно).<br>Оба условия выполнены.<br>Ответ: $\begin{cases} y \le |x^2 - 2| \\ y \ge 0 \end{cases}$ (или $0 \le y \le |x^2 - 2|$).</p><p><strong>в)</strong> На рисунке изображена область, ограниченная гиперболой $y = -\frac{2}{x}$ и осями координат ($x=0$, $y=0$). Все границы сплошные, поэтому неравенства будут нестрогими. Область состоит из двух частей.<br>1. Во второй координатной четверти ($x < 0, y > 0$): область находится между осью $x$ и гиперболой, то есть $0 \le y \le -\frac{2}{x}$. Умножим правую часть неравенства ($y \le -\frac{2}{x}$) на $x$. Так как $x < 0$, знак неравенства меняется на противоположный: $xy \ge -2$. Из левой части ($0 \le y$) при $x < 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.<br>2. В четвертой координатной четверти ($x > 0, y < 0$): область находится между гиперболой и осью $x$, то есть $-\frac{2}{x} \le y \le 0$. Умножим левую часть неравенства ($-\frac{2}{x} \le y$) на $x$. Так как $x > 0$, знак неравенства не меняется: $-2 \le xy$. Из правой части ($y \le 0$) при $x > 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.<br>Так как для обеих частей заштрихованной области получилось одно и то же неравенство, оно и является искомым.<br>Ответ: $-2 \le xy \le 0$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107832" "type" => "task" ] "previous" => array:2 [ "refs" => "1107830" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1163 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Term {#1154 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1152 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:3 [ 0 => App\Models\Term {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1140 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1141 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1145 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1157 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1148} "field_class" => Illuminate\Database\Eloquent\Collection {#1149} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153} "field_author" => Illuminate\Database\Eloquent\Collection {#1150} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1139} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1144} "field_city" => Illuminate\Database\Eloquent\Collection {#1143} "field_series" => Illuminate\Database\Eloquent\Collection {#1155} "field_umk" => Illuminate\Database\Eloquent\Collection {#1162} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1159} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\BookPage {#1199 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097786 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-44" "field_display_title" => "44" "field_folder" => "folder1" "field_image_name" => "44" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1163} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Element {#1212 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1260937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097787" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097785" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1369 #items: array:7 [ 0 => App\Models\Task {#1383 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107830 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-132" "field_display_title" => "132" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1384 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1385 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1387 …2} "content" => Illuminate\Database\Eloquent\Collection {#1397 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1404 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107830 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-132" "field_display_title" => "132" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1384 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1385 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1387 …2} "content" => Illuminate\Database\Eloquent\Collection {#1397 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1404 …2} "page" => array:2 [ …2] ] #changes: 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#dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1399 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107833 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-135" "field_display_title" => "135" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1401 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null 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=> Illuminate\Database\Eloquent\Collection {#1416 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1418 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1414 …2} "content" => Illuminate\Database\Eloquent\Collection {#1413 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1431 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1415 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107834 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-136" "field_display_title" => "136" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1400 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1432 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1428 …2} "content" => Illuminate\Database\Eloquent\Collection {#1427 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1445 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107834 "created_at" => "2026-04-10 13:58:26" "updated_at" => 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#observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1429 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107835 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" …20 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1440 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097786 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-44" "field_display_title" => "44" "field_folder" => "folder1" "field_image_name" => "44" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202} "field_metatags_title" => null "field_metatags_description" => null 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#visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№133 (с. 44)
Условие. №133 (с. 44)
скриншот условия
133. Запишите неравенство, множество решений которого показано на рисунке 18.
а)
$y \ge |x|$
б)
$0 \le y \le |x^2 - 2|$
в)
$-2 \le xy \le 0$
Рисунок 18
Решение. №133 (с. 44)
Решение 2 (rus). №133 (с. 44)
а) На рисунке показана заштрихованная область, которая находится выше графика функции $y = |x|$. Граница области, то есть сам график $y = |x|$, изображена пунктирной линией. Это означает, что точки, лежащие непосредственно на линии, не являются частью решения. Таким образом, для любой точки $(x, y)$ из заштрихованной области должно выполняться строгое неравенство, где значение $y$ больше соответствующего значения на границе. Это приводит к неравенству $y > |x|$.
Для проверки можно взять произвольную точку из заштрихованной области, например, точку $(0, 1)$. Подставив ее координаты в полученное неравенство, получаем $1 > |0|$, или $1 > 0$, что является верным утверждением.
Ответ: $y > |x|$.
б) Заштрихованная область на рисунке ограничена сверху графиком функции $y = |x^2 - 2|$ и снизу осью абсцисс ($y=0$). Обе границы изображены сплошными линиями, что указывает на то, что точки на границах включаются в множество решений. Следовательно, неравенства будут нестрогими.
Условие, что область находится ниже или на графике $y = |x^2 - 2|$, записывается как $y \le |x^2 - 2|$.
Условие, что область находится выше или на оси абсцисс, записывается как $y \ge 0$.
Оба этих условия должны выполняться одновременно, что приводит к системе неравенств. Эту систему можно также записать в виде двойного неравенства.
Проверим на контрольной точке $(0, 1)$ из заштрихованной области:
1) $1 \le |0^2 - 2| \implies 1 \le 2$ (верно).
2) $1 \ge 0$ (верно).
Оба условия выполнены.
Ответ: $\begin{cases} y \le |x^2 - 2| \\ y \ge 0 \end{cases}$ (или $0 \le y \le |x^2 - 2|$).
в) На рисунке изображена область, ограниченная гиперболой $y = -\frac{2}{x}$ и осями координат ($x=0$, $y=0$). Все границы сплошные, поэтому неравенства будут нестрогими. Область состоит из двух частей.
1. Во второй координатной четверти ($x < 0, y > 0$): область находится между осью $x$ и гиперболой, то есть $0 \le y \le -\frac{2}{x}$. Умножим правую часть неравенства ($y \le -\frac{2}{x}$) на $x$. Так как $x < 0$, знак неравенства меняется на противоположный: $xy \ge -2$. Из левой части ($0 \le y$) при $x < 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.
2. В четвертой координатной четверти ($x > 0, y < 0$): область находится между гиперболой и осью $x$, то есть $-\frac{2}{x} \le y \le 0$. Умножим левую часть неравенства ($-\frac{2}{x} \le y$) на $x$. Так как $x > 0$, знак неравенства не меняется: $-2 \le xy$. Из правой части ($y \le 0$) при $x > 0$ следует $xy \le 0$. Итого для этой части: $-2 \le xy \le 0$.
Так как для обеих частей заштрихованной области получилось одно и то же неравенство, оно и является искомым.
Ответ: $-2 \le xy \le 0$.
Другие задания:
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