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Номер 39, страница 37, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 1. Функции, её свойства и график. Параграф 3. Чётные и нечетные функции. Задачи - номер 39, страница 37.
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" => 1119576 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/1-3zad-39" "field_display_title" => "39" "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 {#1072 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 4295 "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 {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 [ …6] …17 } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:4 [ 0 => App\Models\Branch {#1116 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Term {#1117 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Book {#1119 #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" => 4295 "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 {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 {#1122 #items: array:1 [ 0 => App\Models\Term {#1123 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1124 #items: array:1 [ 0 => App\Models\Term {#1125 #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" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #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 {#1126 #items: array:3 [ 0 => App\Models\Term {#1127 #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" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "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 {#1128 #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" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "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 {#1129 #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" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "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 {#1130 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Term {#1132 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 {#1133 #items: array:1 [ 0 => App\Models\Term {#1134 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1137 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1121} "field_class" => Illuminate\Database\Eloquent\Collection {#1122} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1124} "field_author" => Illuminate\Database\Eloquent\Collection {#1126} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1130} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1131} "field_country" => Illuminate\Database\Eloquent\Collection {#1133} "field_city" => Illuminate\Database\Eloquent\Collection {#1135} "field_series" => Illuminate\Database\Eloquent\Collection {#1137} "field_umk" => Illuminate\Database\Eloquent\Collection {#1138} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1139} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1140} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1141} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1147 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1120} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #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 {#1148 #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" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1150 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "10" "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 {#1152 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153} "field_page_start" => "10" "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 {#1152} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151} ] #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\Branch {#1154 #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" => 1119342 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Чётные и нечетные функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1155 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "30" "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 {#1157 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\Branch {#1165 #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" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Term {#1150} ] #escapeWhenCastingToString: false } "field_page_start" => "10" "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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Функции, её свойства и график" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1167} "field_page_start" => "10" "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 {#1168} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169} ] #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" => 1119342 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Чётные и нечетные функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_page_start" => "30" "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 {#1157} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1159 #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" => 1119344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "35" "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 {#1162 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Branch {#1174 #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" => 1119342 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Чётные и нечетные функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Term {#1155} ] #escapeWhenCastingToString: false } "field_page_start" => "30" "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 {#1177 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:1 [ 0 => App\Models\Branch {#1165} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119342 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Чётные и нечетные функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1176} "field_page_start" => "30" "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 {#1177} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1178} ] #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" => 1119344 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_page_start" => "35" "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 {#1162} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1175} ] #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 {#1070 #items: array:2 [ 0 => App\Models\Element {#1097 #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" => 1300261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\Term {#1095 …30} ] #escapeWhenCastingToString: false } "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 {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094} "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 {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "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" => "1119576" "type" => "task" ] "text" => "<p><strong>39.</strong> Решите неравенство.</p><p><strong>a) (1)</strong></p><p>$\frac{-4}{3x-7} > 0;$</p><p><strong>б) (2)</strong></p><p>$\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$</p><p><strong>в) (3)</strong></p><p>$\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}.$ В ответе укажите количество целых решений, принадлежащих отрезку $[-10;12]$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "39-1.jpg" "alt" => null "width" => "1686" "height" => 570 "path" => "/media/algebra_10/baspasy-u19/0-1-3zad/39-1.webp?ts=1753534483" ] ] ] #original: array:7 [ "id" => 1300261 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1119576" "type" => "task" ] "text" => "<p><strong>39.</strong> Решите неравенство.</p><p><strong>a) (1)</strong></p><p>$\frac{-4}{3x-7} > 0;$</p><p><strong>б) (2)</strong></p><p>$\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$</p><p><strong>в) (3)</strong></p><p>$\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}.$ В ответе укажите количество целых решений, принадлежащих отрезку $[-10;12]$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "39-1.jpg" "alt" => null "width" => "1686" "height" => 570 "path" => "/media/algebra_10/baspasy-u19/0-1-3zad/39-1.webp?ts=1753534483" ] ] ] #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 {#1089 #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" => 1551297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1090 #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" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Term {#1088 …30} ] #escapeWhenCastingToString: false } "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 {#1082 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086} "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 {#1082} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "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" => "1119576" "type" => "task" ] "text" => "<p><strong>а) (1)</strong></p> <p>Дано неравенство: $\frac{-4}{3x-7} > 0$.</p> <p>Числитель дроби, $-4$, является отрицательным числом. Чтобы вся дробь была больше нуля (положительной), знаменатель также должен быть отрицательным.</p> <p>Следовательно, решаем неравенство:</p> <p>$3x - 7 < 0$</p> <p>Переносим $-7$ в правую часть, меняя знак:</p> <p>$3x < 7$</p> <p>Делим обе части на 3:</p> <p>$x < \frac{7}{3}$</p> <p>Таким образом, решение неравенства представляет собой интервал от минус бесконечности до $\frac{7}{3}$.</p> <p><strong>Ответ:</strong> $x \in (-\infty; \frac{7}{3})$.</p> <p><strong>б) (2)</strong></p> <p>Дано неравенство: $\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$.</p> <p>Перенесем все члены неравенства в левую часть, чтобы сравнить выражение с нулем:</p> <p>$\frac{5x+4}{5x^2-6x+1} - \frac{1}{x-2} < 0$</p> <p>Разложим на множители знаменатель $5x^2-6x+1$. Для этого найдем корни соответствующего квадратного уравнения $5x^2-6x+1=0$.</p> <p>Дискриминант $D = b^2-4ac = (-6)^2 - 4 \cdot 5 \cdot 1 = 36 - 20 = 16$.</p> <p>Корни уравнения: $x_1 = \frac{-b - \sqrt{D}}{2a} = \frac{6 - \sqrt{16}}{2 \cdot 5} = \frac{6-4}{10} = \frac{2}{10} = \frac{1}{5}$;</p> <p>$x_2 = \frac{-b + \sqrt{D}}{2a} = \frac{6 + \sqrt{16}}{2 \cdot 5} = \frac{6+4}{10} = 1$.</p> <p>Следовательно, $5x^2-6x+1 = 5(x-\frac{1}{5})(x-1) = (5x-1)(x-1)$.</p> <p>Неравенство принимает вид: $\frac{5x+4}{(5x-1)(x-1)} - \frac{1}{x-2} < 0$.</p> <p>Приведем дроби к общему знаменателю $(5x-1)(x-1)(x-2)$:</p> <p>$\frac{(5x+4)(x-2) - 1 \cdot (5x-1)(x-1)}{(5x-1)(x-1)(x-2)} < 0$</p> <p>Раскроем скобки и упростим числитель:</p> <p>$(5x^2 - 10x + 4x - 8) - (5x^2 - 5x - x + 1) = (5x^2 - 6x - 8) - (5x^2 - 6x + 1) = 5x^2 - 6x - 8 - 5x^2 + 6x - 1 = -9$.</p> <p>Неравенство упрощается до: $\frac{-9}{(5x-1)(x-1)(x-2)} < 0$.</p> <p>Так как числитель $-9$ является отрицательным числом, для выполнения неравенства (чтобы вся дробь была меньше нуля) знаменатель должен быть положительным:</p> <p>$(5x-1)(x-1)(x-2) > 0$.</p> <p>Решим это неравенство методом интервалов. Корни (нули) выражения в левой части: $x=\frac{1}{5}$, $x=1$, $x=2$.</p> <p>Эти точки делят числовую ось на четыре интервала. Определим знак выражения на каждом из них:</p> <ul> <li>Интервал $(2; +\infty)$: $x=3 \implies (5\cdot3-1)(3-1)(3-2) = 14 \cdot 2 \cdot 1 > 0$. Знак "+".</li> <li>Интервал $(1; 2)$: $x=1.5 \implies (5\cdot1.5-1)(1.5-1)(1.5-2) = 6.5 \cdot 0.5 \cdot (-0.5) < 0$. Знак "-".</li> <li>Интервал $(\frac{1}{5}; 1)$: $x=0.5 \implies (5\cdot0.5-1)(0.5-1)(0.5-2) = 1.5 \cdot (-0.5) \cdot (-1.5) > 0$. Знак "+".</li> <li>Интервал $(-\infty; \frac{1}{5})$: $x=0 \implies (5\cdot0-1)(0-1)(0-2) = (-1) \cdot (-1) \cdot (-2) < 0$. Знак "-".</li> </ul> <p>Нам нужны интервалы, где выражение больше нуля (знак "+").</p> <p><strong>Ответ:</strong> $x \in (\frac{1}{5}; 1) \cup (2; +\infty)$.</p> <p><strong>в) (3)</strong></p> <p>Дано неравенство: $\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}$.</p> <p>Упростим выражения в числителе и знаменателе. Заметим, что $x^3+6x^2+12x+8$ является формулой куба суммы $(x+2)^3$, а $x^2-2x+1$ — формулой квадрата разности $(x-1)^2$.</p> <p>Подставим эти выражения в неравенство:</p> <p>$\frac{5(x+2)^3}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x-1)^2(x+8)}$</p> <p>Область допустимых значений (ОДЗ): знаменатели не могут быть равны нулю, поэтому $x \neq 1$ и $x \neq -8$.</p> <p>Перенесем правую часть налево:</p> <p>$\frac{5(x+2)^3}{(x-1)^2(x+8)} - \frac{x(x+2)^3}{(x-1)^2(x+8)} \ge 0$</p> <p>Так как знаменатели одинаковы, объединим дроби и вынесем общий множитель $(x+2)^3$ в числителе:</p> <p>$\frac{(x+2)^3(5-x)}{(x-1)^2(x+8)} \ge 0$</p> <p>Решаем неравенство методом интервалов. Найдем нули числителя и знаменателя:</p> <ul> <li>Нули числителя: $x+2=0 \implies x=-2$; $5-x=0 \implies x=5$. Эти точки включаются в решение, так как неравенство нестрогое.</li> <li>Нули знаменателя: $x-1=0 \implies x=1$; $x+8=0 \implies x=-8$. Эти точки исключаются из решения.</li> </ul> <p>Отметим точки $-8, -2, 1, 5$ на числовой оси и определим знаки выражения. Заметим, что множитель $(x-1)^2$ всегда неотрицателен (при $x \neq 1$) и не влияет на знак дроби, поэтому при переходе через точку $x=1$ знак меняться не будет (корень четной кратности).</p> <ul> <li>Интервал $(5; +\infty)$: $x=10 \implies \frac{(+)^3(-)}{(+)^2(+)} = \frac{-}{+} < 0$.</li> <li>Интервал $(1; 5)$: $x=2 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=5$. Получаем $(1; 5]$.</li> <li>Интервал $(-2; 1)$: $x=0 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=-2$. Получаем $[-2; 1)$.</li> <li>Интервал $(-8; -2)$: $x=-3 \implies \frac{(-)^3(+)}{(-)^2(+)} = \frac{-}{+} < 0$.</li> <li>Интервал $(-\infty; -8)$: $x=-10 \implies \frac{(-)^3(+)}{(-)^2(-)} = \frac{-}{-} > 0$. Интервал подходит. Получаем $(-\infty; -8)$.</li> </ul> <p>Общее решение неравенства: $x \in (-\infty; -8) \cup [-2; 1) \cup (1; 5]$.</p> <p>Теперь нужно найти количество целых решений, принадлежащих отрезку $[-10; 12]$.</p> <p>1. Целые числа из интервала $(-\infty; -8)$, принадлежащие отрезку $[-10; 12]$: $-10, -9$ (всего 2 числа).</p> <p>2. Целые числа из объединения $[-2; 1) \cup (1; 5]$, принадлежащие отрезку $[-10; 12]$: $-2, -1, 0, 2, 3, 4, 5$ (всего 7 чисел).</p> <p>Суммируем количество найденных целых решений: $2 + 7 = 9$.</p> <p><strong>Ответ:</strong> 9.</p>" ] #original: array:6 [ "id" => 1551297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1119576" "type" => "task" ] "text" => "<p><strong>а) (1)</strong></p> <p>Дано неравенство: $\frac{-4}{3x-7} > 0$.</p> <p>Числитель дроби, $-4$, является отрицательным числом. Чтобы вся дробь была больше нуля (положительной), знаменатель также должен быть отрицательным.</p> <p>Следовательно, решаем неравенство:</p> <p>$3x - 7 < 0$</p> <p>Переносим $-7$ в правую часть, меняя знак:</p> <p>$3x < 7$</p> <p>Делим обе части на 3:</p> <p>$x < \frac{7}{3}$</p> <p>Таким образом, решение неравенства представляет собой интервал от минус бесконечности до $\frac{7}{3}$.</p> <p><strong>Ответ:</strong> $x \in (-\infty; \frac{7}{3})$.</p> <p><strong>б) (2)</strong></p> <p>Дано неравенство: $\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$.</p> <p>Перенесем все члены неравенства в левую часть, чтобы сравнить выражение с нулем:</p> <p>$\frac{5x+4}{5x^2-6x+1} - \frac{1}{x-2} < 0$</p> <p>Разложим на множители знаменатель $5x^2-6x+1$. Для этого найдем корни соответствующего квадратного уравнения $5x^2-6x+1=0$.</p> <p>Дискриминант $D = b^2-4ac = (-6)^2 - 4 \cdot 5 \cdot 1 = 36 - 20 = 16$.</p> <p>Корни уравнения: $x_1 = \frac{-b - \sqrt{D}}{2a} = \frac{6 - \sqrt{16}}{2 \cdot 5} = \frac{6-4}{10} = \frac{2}{10} = \frac{1}{5}$;</p> <p>$x_2 = \frac{-b + \sqrt{D}}{2a} = \frac{6 + \sqrt{16}}{2 \cdot 5} = \frac{6+4}{10} = 1$.</p> <p>Следовательно, $5x^2-6x+1 = 5(x-\frac{1}{5})(x-1) = (5x-1)(x-1)$.</p> <p>Неравенство принимает вид: $\frac{5x+4}{(5x-1)(x-1)} - \frac{1}{x-2} < 0$.</p> <p>Приведем дроби к общему знаменателю $(5x-1)(x-1)(x-2)$:</p> <p>$\frac{(5x+4)(x-2) - 1 \cdot (5x-1)(x-1)}{(5x-1)(x-1)(x-2)} < 0$</p> <p>Раскроем скобки и упростим числитель:</p> <p>$(5x^2 - 10x + 4x - 8) - (5x^2 - 5x - x + 1) = (5x^2 - 6x - 8) - (5x^2 - 6x + 1) = 5x^2 - 6x - 8 - 5x^2 + 6x - 1 = -9$.</p> <p>Неравенство упрощается до: $\frac{-9}{(5x-1)(x-1)(x-2)} < 0$.</p> <p>Так как числитель $-9$ является отрицательным числом, для выполнения неравенства (чтобы вся дробь была меньше нуля) знаменатель должен быть положительным:</p> <p>$(5x-1)(x-1)(x-2) > 0$.</p> <p>Решим это неравенство методом интервалов. Корни (нули) выражения в левой части: $x=\frac{1}{5}$, $x=1$, $x=2$.</p> <p>Эти точки делят числовую ось на четыре интервала. Определим знак выражения на каждом из них:</p> <ul> <li>Интервал $(2; +\infty)$: $x=3 \implies (5\cdot3-1)(3-1)(3-2) = 14 \cdot 2 \cdot 1 > 0$. Знак "+".</li> <li>Интервал $(1; 2)$: $x=1.5 \implies (5\cdot1.5-1)(1.5-1)(1.5-2) = 6.5 \cdot 0.5 \cdot (-0.5) < 0$. Знак "-".</li> <li>Интервал $(\frac{1}{5}; 1)$: $x=0.5 \implies (5\cdot0.5-1)(0.5-1)(0.5-2) = 1.5 \cdot (-0.5) \cdot (-1.5) > 0$. Знак "+".</li> <li>Интервал $(-\infty; \frac{1}{5})$: $x=0 \implies (5\cdot0-1)(0-1)(0-2) = (-1) \cdot (-1) \cdot (-2) < 0$. Знак "-".</li> </ul> <p>Нам нужны интервалы, где выражение больше нуля (знак "+").</p> <p><strong>Ответ:</strong> $x \in (\frac{1}{5}; 1) \cup (2; +\infty)$.</p> <p><strong>в) (3)</strong></p> <p>Дано неравенство: $\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}$.</p> <p>Упростим выражения в числителе и знаменателе. Заметим, что $x^3+6x^2+12x+8$ является формулой куба суммы $(x+2)^3$, а $x^2-2x+1$ — формулой квадрата разности $(x-1)^2$.</p> <p>Подставим эти выражения в неравенство:</p> <p>$\frac{5(x+2)^3}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x-1)^2(x+8)}$</p> <p>Область допустимых значений (ОДЗ): знаменатели не могут быть равны нулю, поэтому $x \neq 1$ и $x \neq -8$.</p> <p>Перенесем правую часть налево:</p> <p>$\frac{5(x+2)^3}{(x-1)^2(x+8)} - \frac{x(x+2)^3}{(x-1)^2(x+8)} \ge 0$</p> <p>Так как знаменатели одинаковы, объединим дроби и вынесем общий множитель $(x+2)^3$ в числителе:</p> <p>$\frac{(x+2)^3(5-x)}{(x-1)^2(x+8)} \ge 0$</p> <p>Решаем неравенство методом интервалов. Найдем нули числителя и знаменателя:</p> <ul> <li>Нули числителя: $x+2=0 \implies x=-2$; $5-x=0 \implies x=5$. Эти точки включаются в решение, так как неравенство нестрогое.</li> <li>Нули знаменателя: $x-1=0 \implies x=1$; $x+8=0 \implies x=-8$. Эти точки исключаются из решения.</li> </ul> <p>Отметим точки $-8, -2, 1, 5$ на числовой оси и определим знаки выражения. Заметим, что множитель $(x-1)^2$ всегда неотрицателен (при $x \neq 1$) и не влияет на знак дроби, поэтому при переходе через точку $x=1$ знак меняться не будет (корень четной кратности).</p> <ul> <li>Интервал $(5; +\infty)$: $x=10 \implies \frac{(+)^3(-)}{(+)^2(+)} = \frac{-}{+} < 0$.</li> <li>Интервал $(1; 5)$: $x=2 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=5$. Получаем $(1; 5]$.</li> <li>Интервал $(-2; 1)$: $x=0 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=-2$. Получаем $[-2; 1)$.</li> <li>Интервал $(-8; -2)$: $x=-3 \implies \frac{(-)^3(+)}{(-)^2(+)} = \frac{-}{+} < 0$.</li> <li>Интервал $(-\infty; -8)$: $x=-10 \implies \frac{(-)^3(+)}{(-)^2(-)} = \frac{-}{-} > 0$. Интервал подходит. Получаем $(-\infty; -8)$.</li> </ul> <p>Общее решение неравенства: $x \in (-\infty; -8) \cup [-2; 1) \cup (1; 5]$.</p> <p>Теперь нужно найти количество целых решений, принадлежащих отрезку $[-10; 12]$.</p> <p>1. Целые числа из интервала $(-\infty; -8)$, принадлежащие отрезку $[-10; 12]$: $-10, -9$ (всего 2 числа).</p> <p>2. Целые числа из объединения $[-2; 1) \cup (1; 5]$, принадлежащие отрезку $[-10; 12]$: $-2, -1, 0, 2, 3, 4, 5$ (всего 7 чисел).</p> <p>Суммируем количество найденных целых решений: $2 + 7 = 9$.</p> <p><strong>Ответ:</strong> 9.</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" => "1119577" "type" => "task" ] "previous" => array:2 [ "refs" => "1119575" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\BookPage {#1101 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" 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№39 (с. 37)
Условие. №39 (с. 37)
скриншот условия
39. Решите неравенство.
a) (1)
$\frac{-4}{3x-7} > 0;$
б) (2)
$\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$
в) (3)
$\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}.$ В ответе укажите количество целых решений, принадлежащих отрезку $[-10;12]$.
Решение 2 (rus). №39 (с. 37)
а) (1)
Дано неравенство: $\frac{-4}{3x-7} > 0$.
Числитель дроби, $-4$, является отрицательным числом. Чтобы вся дробь была больше нуля (положительной), знаменатель также должен быть отрицательным.
Следовательно, решаем неравенство:
$3x - 7 < 0$
Переносим $-7$ в правую часть, меняя знак:
$3x < 7$
Делим обе части на 3:
$x < \frac{7}{3}$
Таким образом, решение неравенства представляет собой интервал от минус бесконечности до $\frac{7}{3}$.
Ответ: $x \in (-\infty; \frac{7}{3})$.
б) (2)
Дано неравенство: $\frac{5x+4}{5x^2-6x+1} < \frac{1}{x-2}$.
Перенесем все члены неравенства в левую часть, чтобы сравнить выражение с нулем:
$\frac{5x+4}{5x^2-6x+1} - \frac{1}{x-2} < 0$
Разложим на множители знаменатель $5x^2-6x+1$. Для этого найдем корни соответствующего квадратного уравнения $5x^2-6x+1=0$.
Дискриминант $D = b^2-4ac = (-6)^2 - 4 \cdot 5 \cdot 1 = 36 - 20 = 16$.
Корни уравнения: $x_1 = \frac{-b - \sqrt{D}}{2a} = \frac{6 - \sqrt{16}}{2 \cdot 5} = \frac{6-4}{10} = \frac{2}{10} = \frac{1}{5}$;
$x_2 = \frac{-b + \sqrt{D}}{2a} = \frac{6 + \sqrt{16}}{2 \cdot 5} = \frac{6+4}{10} = 1$.
Следовательно, $5x^2-6x+1 = 5(x-\frac{1}{5})(x-1) = (5x-1)(x-1)$.
Неравенство принимает вид: $\frac{5x+4}{(5x-1)(x-1)} - \frac{1}{x-2} < 0$.
Приведем дроби к общему знаменателю $(5x-1)(x-1)(x-2)$:
$\frac{(5x+4)(x-2) - 1 \cdot (5x-1)(x-1)}{(5x-1)(x-1)(x-2)} < 0$
Раскроем скобки и упростим числитель:
$(5x^2 - 10x + 4x - 8) - (5x^2 - 5x - x + 1) = (5x^2 - 6x - 8) - (5x^2 - 6x + 1) = 5x^2 - 6x - 8 - 5x^2 + 6x - 1 = -9$.
Неравенство упрощается до: $\frac{-9}{(5x-1)(x-1)(x-2)} < 0$.
Так как числитель $-9$ является отрицательным числом, для выполнения неравенства (чтобы вся дробь была меньше нуля) знаменатель должен быть положительным:
$(5x-1)(x-1)(x-2) > 0$.
Решим это неравенство методом интервалов. Корни (нули) выражения в левой части: $x=\frac{1}{5}$, $x=1$, $x=2$.
Эти точки делят числовую ось на четыре интервала. Определим знак выражения на каждом из них:
- Интервал $(2; +\infty)$: $x=3 \implies (5\cdot3-1)(3-1)(3-2) = 14 \cdot 2 \cdot 1 > 0$. Знак "+".
- Интервал $(1; 2)$: $x=1.5 \implies (5\cdot1.5-1)(1.5-1)(1.5-2) = 6.5 \cdot 0.5 \cdot (-0.5) < 0$. Знак "-".
- Интервал $(\frac{1}{5}; 1)$: $x=0.5 \implies (5\cdot0.5-1)(0.5-1)(0.5-2) = 1.5 \cdot (-0.5) \cdot (-1.5) > 0$. Знак "+".
- Интервал $(-\infty; \frac{1}{5})$: $x=0 \implies (5\cdot0-1)(0-1)(0-2) = (-1) \cdot (-1) \cdot (-2) < 0$. Знак "-".
Нам нужны интервалы, где выражение больше нуля (знак "+").
Ответ: $x \in (\frac{1}{5}; 1) \cup (2; +\infty)$.
в) (3)
Дано неравенство: $\frac{5(x^3+6x^2+12x+8)}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x^2-2x+1)(x+8)}$.
Упростим выражения в числителе и знаменателе. Заметим, что $x^3+6x^2+12x+8$ является формулой куба суммы $(x+2)^3$, а $x^2-2x+1$ — формулой квадрата разности $(x-1)^2$.
Подставим эти выражения в неравенство:
$\frac{5(x+2)^3}{(x-1)^2(x+8)} \ge \frac{x(x+2)^3}{(x-1)^2(x+8)}$
Область допустимых значений (ОДЗ): знаменатели не могут быть равны нулю, поэтому $x \neq 1$ и $x \neq -8$.
Перенесем правую часть налево:
$\frac{5(x+2)^3}{(x-1)^2(x+8)} - \frac{x(x+2)^3}{(x-1)^2(x+8)} \ge 0$
Так как знаменатели одинаковы, объединим дроби и вынесем общий множитель $(x+2)^3$ в числителе:
$\frac{(x+2)^3(5-x)}{(x-1)^2(x+8)} \ge 0$
Решаем неравенство методом интервалов. Найдем нули числителя и знаменателя:
- Нули числителя: $x+2=0 \implies x=-2$; $5-x=0 \implies x=5$. Эти точки включаются в решение, так как неравенство нестрогое.
- Нули знаменателя: $x-1=0 \implies x=1$; $x+8=0 \implies x=-8$. Эти точки исключаются из решения.
Отметим точки $-8, -2, 1, 5$ на числовой оси и определим знаки выражения. Заметим, что множитель $(x-1)^2$ всегда неотрицателен (при $x \neq 1$) и не влияет на знак дроби, поэтому при переходе через точку $x=1$ знак меняться не будет (корень четной кратности).
- Интервал $(5; +\infty)$: $x=10 \implies \frac{(+)^3(-)}{(+)^2(+)} = \frac{-}{+} < 0$.
- Интервал $(1; 5)$: $x=2 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=5$. Получаем $(1; 5]$.
- Интервал $(-2; 1)$: $x=0 \implies \frac{(+)^3(+)}{(+)^2(+)} = \frac{+}{+} > 0$. Интервал подходит. Включаем $x=-2$. Получаем $[-2; 1)$.
- Интервал $(-8; -2)$: $x=-3 \implies \frac{(-)^3(+)}{(-)^2(+)} = \frac{-}{+} < 0$.
- Интервал $(-\infty; -8)$: $x=-10 \implies \frac{(-)^3(+)}{(-)^2(-)} = \frac{-}{-} > 0$. Интервал подходит. Получаем $(-\infty; -8)$.
Общее решение неравенства: $x \in (-\infty; -8) \cup [-2; 1) \cup (1; 5]$.
Теперь нужно найти количество целых решений, принадлежащих отрезку $[-10; 12]$.
1. Целые числа из интервала $(-\infty; -8)$, принадлежащие отрезку $[-10; 12]$: $-10, -9$ (всего 2 числа).
2. Целые числа из объединения $[-2; 1) \cup (1; 5]$, принадлежащие отрезку $[-10; 12]$: $-2, -1, 0, 2, 3, 4, 5$ (всего 7 чисел).
Суммируем количество найденных целых решений: $2 + 7 = 9$.
Ответ: 9.
Другие задания:
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