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Номер 179, страница 55 - 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. 6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы» - номер 179, страница 55.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #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" => 1107880 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-179" "field_display_title" => "179" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #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_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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#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" => 1107645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "54" "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" => 1107645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "54" "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 {#1141 #items: array:3 [ 0 => App\Models\Element {#1150 #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" => 1276534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Edition {#1151 #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 {#1153 …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 {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …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 {#1153 …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 {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 …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" => "1107880" "type" => "task" ] "text" => "<p><strong>179.</strong> Постройте в координатной плоскости фигуру, координаты каждой точки которой являются решениями системы неравенств:</p><p><strong>a)</strong> $\begin{cases} x + y \le 2, \\ x + 2 \ge 0; \end{cases}$</p><p><strong>б)</strong> $\begin{cases} y - 3x - 4 \le 0, \\ 2y - 6x + 8 \ge 0; \end{cases}$</p><p><strong>в)</strong> $\begin{cases} y < \frac{4}{x}, \\ y \ge x^2; \end{cases}$</p><p><strong>г)</strong> $\begin{cases} |x| < 2, \\ x^2 + y^2 < 9. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "179-1.jpg" "alt" => null "width" => "1554" "height" => 569 "path" => "/media/algebra_09/soltan-u19/0-1/179-1.webp?ts=1752834240" ] ] ] #original: array:7 [ "id" => 1276534 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "1107880" "type" => "task" ] "text" => "<p><strong>179.</strong> Постройте в координатной плоскости фигуру, координаты каждой точки которой являются решениями системы неравенств:</p><p><strong>a)</strong> $\begin{cases} x + y \le 2, \\ x + 2 \ge 0; \end{cases}$</p><p><strong>б)</strong> $\begin{cases} y - 3x - 4 \le 0, \\ 2y - 6x + 8 \ge 0; \end{cases}$</p><p><strong>в)</strong> $\begin{cases} y < \frac{4}{x}, \\ y \ge x^2; \end{cases}$</p><p><strong>г)</strong> $\begin{cases} |x| < 2, \\ x^2 + y^2 < 9. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "179-1.jpg" "alt" => null "width" => "1554" "height" => 569 "path" => "/media/algebra_09/soltan-u19/0-1/179-1.webp?ts=1752834240" ] ] ] #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 {#1157 #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" => 1277825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1158 #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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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" => "1107880" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "179-1.jpg" "alt" => null "width" => "1438" "height" => 1608 "path" => "/media/algebra_09/soltan-u19/1-1/179-1.webp?ts=1752830923" ] ] ] #original: array:6 [ "id" => 1277825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1107880" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "179-1.jpg" "alt" => null "width" => "1438" "height" => 1608 "path" => "/media/algebra_09/soltan-u19/1-1/179-1.webp?ts=1752830923" ] ] ] #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 {#1165 #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" => 1553780 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1169 …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 {#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" => 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 {#1169 …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 {#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" => "1107880" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} x + y \le 2 \\ x + 2 \ge 0 \end{cases} $</p><p>Преобразуем каждое неравенство, чтобы выразить $y$ или $x$:</p><p><strong>1.</strong> Из первого неравенства $x + y \le 2$ получаем $y \le -x + 2$. Это неравенство описывает все точки на координатной плоскости, которые лежат на прямой $y = -x + 2$ и ниже нее. Граничная прямая $y = -x + 2$ строится по двум точкам, например, $(0, 2)$ и $(2, 0)$. Так как неравенство нестрогое ($\le$), прямая является частью решения и изображается сплошной линией.</p><p><strong>2.</strong> Из второго неравенства $x + 2 \ge 0$ получаем $x \ge -2$. Это неравенство описывает все точки, которые лежат на вертикальной прямой $x = -2$ и правее нее. Так как неравенство нестрогое ($\ge$), эта прямая также является частью решения и изображается сплошной линией.</p><p>Решением системы является пересечение (общая область) этих двух полуплоскостей. Чтобы найти вершину полученной угловой области, найдем точку пересечения граничных прямых $y = -x + 2$ и $x = -2$. Подставив $x = -2$ в уравнение первой прямой, получим $y = -(-2) + 2 = 4$. Таким образом, вершина находится в точке $(-2, 4)$.</p><p><strong>Ответ:</strong> Фигура, являющаяся решением системы, — это неограниченная область на плоскости (угол), ограниченная слева прямой $x = -2$ и сверху прямой $y = -x + 2$. Границы области (сами прямые) включены в решение. Вершина угла находится в точке $(-2, 4)$.</p><p><strong>б)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} y - 3x - 4 \le 0 \\ 2y - 6x + 8 \ge 0 \end{cases} $</p><p>Преобразуем оба неравенства:</p><p><strong>1.</strong> $y - 3x - 4 \le 0 \implies y \le 3x + 4$. Это полуплоскость, лежащая на прямой $y = 3x + 4$ и ниже нее.</p><p><strong>2.</strong> $2y - 6x + 8 \ge 0$. Разделим все члены на 2: $y - 3x + 4 \ge 0 \implies y \ge 3x - 4$. Это полуплоскость, лежащая на прямой $y = 3x - 4$ и выше нее.</p><p>Граничные прямые $y = 3x + 4$ и $y = 3x - 4$ имеют одинаковый угловой коэффициент $k=3$, что означает, что они параллельны.</p><p>Решением системы является пересечение двух полуплоскостей, то есть область, заключенная между этими двумя параллельными прямыми. Поскольку оба неравенства нестрогие ($\le$ и $\ge$), сами граничные прямые включаются в решение.</p><p><strong>Ответ:</strong> Фигура представляет собой бесконечную полосу, заключенную между параллельными прямыми $y = 3x + 4$ и $y = 3x - 4$. Границы полосы включены в решение.</p><p><strong>в)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} y < \frac{4}{x} \\ y \ge x^2 \end{cases} $</p><p><strong>1.</strong> Первое неравенство $y < \frac{4}{x}$ задает область под графиком функции $y = \frac{4}{x}$ (гипербола). Так как неравенство строгое ($<$), точки на самой гиперболе не являются частью решения, и она изображается пунктирной линией. Отметим, что $x \ne 0$.</p><p><strong>2.</strong> Второе неравенство $y \ge x^2$ задает область на параболе $y = x^2$ и выше нее. Так как неравенство нестрогое ($\ge$), сама парабола включена в решение и изображается сплошной линией.</p><p>Решением системы являются точки $(x, y)$, которые удовлетворяют обоим условиям, то есть $x^2 \le y < \frac{4}{x}$. Это возможно только в случае, если $x^2 < \frac{4}{x}$.</p><p>- При $x > 0$ умножаем на $x$, получаем $x^3 < 4$, откуда $0 < x < \sqrt[3]{4}$.</p><p>- При $x < 0$ умножаем на $x$ и меняем знак неравенства: $x^3 > 4$. Это неравенство не имеет решений для отрицательных $x$, так как куб отрицательного числа всегда отрицателен.</p><p>Таким образом, искомая фигура существует только в первой координатной четверти, где $0 < x < \sqrt[3]{4}$. Она ограничена снизу параболой $y=x^2$ и сверху гиперболой $y=4/x$. Найдем точку пересечения границ: $x^2 = \frac{4}{x} \implies x^3 = 4 \implies x = \sqrt[3]{4}$. Координата $y$ в этой точке: $y = (\sqrt[3]{4})^2 = \sqrt[3]{16}$.</p><p><strong>Ответ:</strong> Фигура является конечной областью в первой координатной четверти. Она ограничена снизу дугой параболы $y=x^2$ (включительно), сверху — дугой гиперболы $y=4/x$ (не включительно). Слева область неявно ограничена осью $Oy$ (не включительно). Область существует в интервале $0 < x \le \sqrt[3]{4}$.</p><p><strong>г)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} |x| < 2 \\ x^2 + y^2 < 9 \end{cases} $</p><p><strong>1.</strong> Неравенство $|x| < 2$ эквивалентно $-2 < x < 2$. Оно задает открытую вертикальную полосу между прямыми $x = -2$ и $x = 2$. Сами прямые не входят в решение.</p><p><strong>2.</strong> Неравенство $x^2 + y^2 < 9$ (или $x^2 + y^2 < 3^2$) задает множество точек внутри круга с центром в начале координат $(0, 0)$ и радиусом 3. Сама окружность не входит в решение.</p><p>Решением системы является пересечение этих двух областей — это центральная часть круга, вырезанная вертикальной полосой. Поскольку все неравенства строгие, границы полученной фигуры не включаются в решение.</p><p>Найдем точки пересечения граничных линий $x^2+y^2=9$ с прямыми $x=\pm 2$:</p><p>При $x = \pm 2$, имеем $(\pm 2)^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \implies y = \pm\sqrt{5}$.</p><p>Фигура ограничена двумя вертикальными отрезками (на прямых $x=-2$ и $x=2$ между $y=-\sqrt{5}$ и $y=\sqrt{5}$) и двумя дугами окружности, соединяющими концы этих отрезков.</p><p><strong>Ответ:</strong> Фигура представляет собой открытый сегмент круга: внутренность круга с центром в $(0,0)$ и радиусом 3, находящаяся в полосе $-2 < x < 2$. Границы фигуры (дуги окружности и вертикальные отрезки) не включены в решение.</p>" ] #original: array:6 [ "id" => 1553780 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1107880" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} x + y \le 2 \\ x + 2 \ge 0 \end{cases} $</p><p>Преобразуем каждое неравенство, чтобы выразить $y$ или $x$:</p><p><strong>1.</strong> Из первого неравенства $x + y \le 2$ получаем $y \le -x + 2$. Это неравенство описывает все точки на координатной плоскости, которые лежат на прямой $y = -x + 2$ и ниже нее. Граничная прямая $y = -x + 2$ строится по двум точкам, например, $(0, 2)$ и $(2, 0)$. Так как неравенство нестрогое ($\le$), прямая является частью решения и изображается сплошной линией.</p><p><strong>2.</strong> Из второго неравенства $x + 2 \ge 0$ получаем $x \ge -2$. Это неравенство описывает все точки, которые лежат на вертикальной прямой $x = -2$ и правее нее. Так как неравенство нестрогое ($\ge$), эта прямая также является частью решения и изображается сплошной линией.</p><p>Решением системы является пересечение (общая область) этих двух полуплоскостей. Чтобы найти вершину полученной угловой области, найдем точку пересечения граничных прямых $y = -x + 2$ и $x = -2$. Подставив $x = -2$ в уравнение первой прямой, получим $y = -(-2) + 2 = 4$. Таким образом, вершина находится в точке $(-2, 4)$.</p><p><strong>Ответ:</strong> Фигура, являющаяся решением системы, — это неограниченная область на плоскости (угол), ограниченная слева прямой $x = -2$ и сверху прямой $y = -x + 2$. Границы области (сами прямые) включены в решение. Вершина угла находится в точке $(-2, 4)$.</p><p><strong>б)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} y - 3x - 4 \le 0 \\ 2y - 6x + 8 \ge 0 \end{cases} $</p><p>Преобразуем оба неравенства:</p><p><strong>1.</strong> $y - 3x - 4 \le 0 \implies y \le 3x + 4$. Это полуплоскость, лежащая на прямой $y = 3x + 4$ и ниже нее.</p><p><strong>2.</strong> $2y - 6x + 8 \ge 0$. Разделим все члены на 2: $y - 3x + 4 \ge 0 \implies y \ge 3x - 4$. Это полуплоскость, лежащая на прямой $y = 3x - 4$ и выше нее.</p><p>Граничные прямые $y = 3x + 4$ и $y = 3x - 4$ имеют одинаковый угловой коэффициент $k=3$, что означает, что они параллельны.</p><p>Решением системы является пересечение двух полуплоскостей, то есть область, заключенная между этими двумя параллельными прямыми. Поскольку оба неравенства нестрогие ($\le$ и $\ge$), сами граничные прямые включаются в решение.</p><p><strong>Ответ:</strong> Фигура представляет собой бесконечную полосу, заключенную между параллельными прямыми $y = 3x + 4$ и $y = 3x - 4$. Границы полосы включены в решение.</p><p><strong>в)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} y < \frac{4}{x} \\ y \ge x^2 \end{cases} $</p><p><strong>1.</strong> Первое неравенство $y < \frac{4}{x}$ задает область под графиком функции $y = \frac{4}{x}$ (гипербола). Так как неравенство строгое ($<$), точки на самой гиперболе не являются частью решения, и она изображается пунктирной линией. Отметим, что $x \ne 0$.</p><p><strong>2.</strong> Второе неравенство $y \ge x^2$ задает область на параболе $y = x^2$ и выше нее. Так как неравенство нестрогое ($\ge$), сама парабола включена в решение и изображается сплошной линией.</p><p>Решением системы являются точки $(x, y)$, которые удовлетворяют обоим условиям, то есть $x^2 \le y < \frac{4}{x}$. Это возможно только в случае, если $x^2 < \frac{4}{x}$.</p><p>- При $x > 0$ умножаем на $x$, получаем $x^3 < 4$, откуда $0 < x < \sqrt[3]{4}$.</p><p>- При $x < 0$ умножаем на $x$ и меняем знак неравенства: $x^3 > 4$. Это неравенство не имеет решений для отрицательных $x$, так как куб отрицательного числа всегда отрицателен.</p><p>Таким образом, искомая фигура существует только в первой координатной четверти, где $0 < x < \sqrt[3]{4}$. Она ограничена снизу параболой $y=x^2$ и сверху гиперболой $y=4/x$. Найдем точку пересечения границ: $x^2 = \frac{4}{x} \implies x^3 = 4 \implies x = \sqrt[3]{4}$. Координата $y$ в этой точке: $y = (\sqrt[3]{4})^2 = \sqrt[3]{16}$.</p><p><strong>Ответ:</strong> Фигура является конечной областью в первой координатной четверти. Она ограничена снизу дугой параболы $y=x^2$ (включительно), сверху — дугой гиперболы $y=4/x$ (не включительно). Слева область неявно ограничена осью $Oy$ (не включительно). Область существует в интервале $0 < x \le \sqrt[3]{4}$.</p><p><strong>г)</strong></p><p>Рассмотрим систему неравенств:</p><p>$ \begin{cases} |x| < 2 \\ x^2 + y^2 < 9 \end{cases} $</p><p><strong>1.</strong> Неравенство $|x| < 2$ эквивалентно $-2 < x < 2$. Оно задает открытую вертикальную полосу между прямыми $x = -2$ и $x = 2$. Сами прямые не входят в решение.</p><p><strong>2.</strong> Неравенство $x^2 + y^2 < 9$ (или $x^2 + y^2 < 3^2$) задает множество точек внутри круга с центром в начале координат $(0, 0)$ и радиусом 3. Сама окружность не входит в решение.</p><p>Решением системы является пересечение этих двух областей — это центральная часть круга, вырезанная вертикальной полосой. Поскольку все неравенства строгие, границы полученной фигуры не включаются в решение.</p><p>Найдем точки пересечения граничных линий $x^2+y^2=9$ с прямыми $x=\pm 2$:</p><p>При $x = \pm 2$, имеем $(\pm 2)^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \implies y = \pm\sqrt{5}$.</p><p>Фигура ограничена двумя вертикальными отрезками (на прямых $x=-2$ и $x=2$ между $y=-\sqrt{5}$ и $y=\sqrt{5}$) и двумя дугами окружности, соединяющими концы этих отрезков.</p><p><strong>Ответ:</strong> Фигура представляет собой открытый сегмент круга: внутренность круга с центром в $(0,0)$ и радиусом 3, находящаяся в полосе $-2 < x < 2$. Границы фигуры (дуги окружности и вертикальные отрезки) не включены в решение.</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" => "1107881" "type" => "task" ] "previous" => array:2 [ "refs" => "1107879" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1147 #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" 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…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 {#1145 #items: array:3 [ 0 => App\Models\Term {#1143 #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 {#1173 #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 {#1174 #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 {#1176 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Term {#1178 #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 {#1179 #items: array:1 [ 0 => App\Models\Term {#1180 #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 {#1181 #items: array:1 [ 0 => App\Models\Term {#1182 #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 {#1183 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #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 {#1140} "field_class" => Illuminate\Database\Eloquent\Collection {#1146} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1144} "field_author" => Illuminate\Database\Eloquent\Collection {#1145} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1176} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_country" => Illuminate\Database\Eloquent\Collection {#1179} "field_city" => Illuminate\Database\Eloquent\Collection {#1181} "field_series" => Illuminate\Database\Eloquent\Collection {#1183} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "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" => 1097797 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-55" "field_display_title" => "55" "field_folder" => "folder1" "field_image_name" => "55" "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 {#1147} ] #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" => 1260948 "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" => 1260948 "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" => "1097798" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097796" "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" => 1107875 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-174" "field_display_title" => "174" "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 {#1390 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1396 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107875 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-174" "field_display_title" => "174" "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 {#1390 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1396 …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 => "*" ] } 1 => App\Models\Task {#1389 #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" => 1107876 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-175" "field_display_title" => "175" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1388 …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 {#1395 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1393 …2} "content" => Illuminate\Database\Eloquent\Collection {#1411 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1418 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107876 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-175" "field_display_title" => "175" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1388 …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 {#1395 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1393 …2} "content" => Illuminate\Database\Eloquent\Collection {#1411 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1418 …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 => "*" ] } 2 => App\Models\Task {#1391 #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" => 1107877 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-176" "field_display_title" => "176" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1392 …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 {#1394 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1409 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408 …2} "content" => Illuminate\Database\Eloquent\Collection {#1406 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1413 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107877 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "55" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-176" "field_display_title" => "176" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1392 …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 {#1394 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1409 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408 …2} "content" => Illuminate\Database\Eloquent\Collection {#1406 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1413 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] 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№179 (с. 55)
Условие. №179 (с. 55)
скриншот условия
179. Постройте в координатной плоскости фигуру, координаты каждой точки которой являются решениями системы неравенств:
a) $\begin{cases} x + y \le 2, \\ x + 2 \ge 0; \end{cases}$
б) $\begin{cases} y - 3x - 4 \le 0, \\ 2y - 6x + 8 \ge 0; \end{cases}$
в) $\begin{cases} y < \frac{4}{x}, \\ y \ge x^2; \end{cases}$
г) $\begin{cases} |x| < 2, \\ x^2 + y^2 < 9. \end{cases}$
Решение. №179 (с. 55)
Решение 2 (rus). №179 (с. 55)
а)
Рассмотрим систему неравенств:
$ \begin{cases} x + y \le 2 \\ x + 2 \ge 0 \end{cases} $
Преобразуем каждое неравенство, чтобы выразить $y$ или $x$:
1. Из первого неравенства $x + y \le 2$ получаем $y \le -x + 2$. Это неравенство описывает все точки на координатной плоскости, которые лежат на прямой $y = -x + 2$ и ниже нее. Граничная прямая $y = -x + 2$ строится по двум точкам, например, $(0, 2)$ и $(2, 0)$. Так как неравенство нестрогое ($\le$), прямая является частью решения и изображается сплошной линией.
2. Из второго неравенства $x + 2 \ge 0$ получаем $x \ge -2$. Это неравенство описывает все точки, которые лежат на вертикальной прямой $x = -2$ и правее нее. Так как неравенство нестрогое ($\ge$), эта прямая также является частью решения и изображается сплошной линией.
Решением системы является пересечение (общая область) этих двух полуплоскостей. Чтобы найти вершину полученной угловой области, найдем точку пересечения граничных прямых $y = -x + 2$ и $x = -2$. Подставив $x = -2$ в уравнение первой прямой, получим $y = -(-2) + 2 = 4$. Таким образом, вершина находится в точке $(-2, 4)$.
Ответ: Фигура, являющаяся решением системы, — это неограниченная область на плоскости (угол), ограниченная слева прямой $x = -2$ и сверху прямой $y = -x + 2$. Границы области (сами прямые) включены в решение. Вершина угла находится в точке $(-2, 4)$.
б)
Рассмотрим систему неравенств:
$ \begin{cases} y - 3x - 4 \le 0 \\ 2y - 6x + 8 \ge 0 \end{cases} $
Преобразуем оба неравенства:
1. $y - 3x - 4 \le 0 \implies y \le 3x + 4$. Это полуплоскость, лежащая на прямой $y = 3x + 4$ и ниже нее.
2. $2y - 6x + 8 \ge 0$. Разделим все члены на 2: $y - 3x + 4 \ge 0 \implies y \ge 3x - 4$. Это полуплоскость, лежащая на прямой $y = 3x - 4$ и выше нее.
Граничные прямые $y = 3x + 4$ и $y = 3x - 4$ имеют одинаковый угловой коэффициент $k=3$, что означает, что они параллельны.
Решением системы является пересечение двух полуплоскостей, то есть область, заключенная между этими двумя параллельными прямыми. Поскольку оба неравенства нестрогие ($\le$ и $\ge$), сами граничные прямые включаются в решение.
Ответ: Фигура представляет собой бесконечную полосу, заключенную между параллельными прямыми $y = 3x + 4$ и $y = 3x - 4$. Границы полосы включены в решение.
в)
Рассмотрим систему неравенств:
$ \begin{cases} y < \frac{4}{x} \\ y \ge x^2 \end{cases} $
1. Первое неравенство $y < \frac{4}{x}$ задает область под графиком функции $y = \frac{4}{x}$ (гипербола). Так как неравенство строгое ($<$), точки на самой гиперболе не являются частью решения, и она изображается пунктирной линией. Отметим, что $x \ne 0$.
2. Второе неравенство $y \ge x^2$ задает область на параболе $y = x^2$ и выше нее. Так как неравенство нестрогое ($\ge$), сама парабола включена в решение и изображается сплошной линией.
Решением системы являются точки $(x, y)$, которые удовлетворяют обоим условиям, то есть $x^2 \le y < \frac{4}{x}$. Это возможно только в случае, если $x^2 < \frac{4}{x}$.
- При $x > 0$ умножаем на $x$, получаем $x^3 < 4$, откуда $0 < x < \sqrt[3]{4}$.
- При $x < 0$ умножаем на $x$ и меняем знак неравенства: $x^3 > 4$. Это неравенство не имеет решений для отрицательных $x$, так как куб отрицательного числа всегда отрицателен.
Таким образом, искомая фигура существует только в первой координатной четверти, где $0 < x < \sqrt[3]{4}$. Она ограничена снизу параболой $y=x^2$ и сверху гиперболой $y=4/x$. Найдем точку пересечения границ: $x^2 = \frac{4}{x} \implies x^3 = 4 \implies x = \sqrt[3]{4}$. Координата $y$ в этой точке: $y = (\sqrt[3]{4})^2 = \sqrt[3]{16}$.
Ответ: Фигура является конечной областью в первой координатной четверти. Она ограничена снизу дугой параболы $y=x^2$ (включительно), сверху — дугой гиперболы $y=4/x$ (не включительно). Слева область неявно ограничена осью $Oy$ (не включительно). Область существует в интервале $0 < x \le \sqrt[3]{4}$.
г)
Рассмотрим систему неравенств:
$ \begin{cases} |x| < 2 \\ x^2 + y^2 < 9 \end{cases} $
1. Неравенство $|x| < 2$ эквивалентно $-2 < x < 2$. Оно задает открытую вертикальную полосу между прямыми $x = -2$ и $x = 2$. Сами прямые не входят в решение.
2. Неравенство $x^2 + y^2 < 9$ (или $x^2 + y^2 < 3^2$) задает множество точек внутри круга с центром в начале координат $(0, 0)$ и радиусом 3. Сама окружность не входит в решение.
Решением системы является пересечение этих двух областей — это центральная часть круга, вырезанная вертикальной полосой. Поскольку все неравенства строгие, границы полученной фигуры не включаются в решение.
Найдем точки пересечения граничных линий $x^2+y^2=9$ с прямыми $x=\pm 2$:
При $x = \pm 2$, имеем $(\pm 2)^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \implies y = \pm\sqrt{5}$.
Фигура ограничена двумя вертикальными отрезками (на прямых $x=-2$ и $x=2$ между $y=-\sqrt{5}$ и $y=\sqrt{5}$) и двумя дугами окружности, соединяющими концы этих отрезков.
Ответ: Фигура представляет собой открытый сегмент круга: внутренность круга с центром в $(0,0)$ и радиусом 3, находящаяся в полосе $-2 < x < 2$. Границы фигуры (дуги окружности и вертикальные отрезки) не включены в решение.
Другие задания:
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