{"id":156214,"date":"2024-10-17T05:33:19","date_gmt":"2024-10-17T05:33:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=156214"},"modified":"2024-10-17T05:33:22","modified_gmt":"2024-10-17T05:33:22","slug":"a-radio-station-is-giving-away-tickets-to-a-play-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/17\/a-radio-station-is-giving-away-tickets-to-a-play-2\/","title":{"rendered":"A radio station is giving away tickets to a play."},"content":{"rendered":"\n<p>A radio station is giving away tickets to a play. They plan to give away tickets for seats that cost $10 and $20. They want to give away at least 20 tickets. The total cost of all the tickets they give away can be no more than $280. Write the inequalities that represent this situation and graph.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this problem, we need to create a system of inequalities based on the given conditions. Let\u2019s define the variables:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Let <strong>x<\/strong> represent the number of $10 tickets.<\/li>\n\n\n\n<li>Let <strong>y<\/strong> represent the number of $20 tickets.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Creating the inequalities<\/h3>\n\n\n\n<p>The radio station is giving away at least 20 tickets, which leads to the inequality:<br>[ x + y \\geq 20 ]<br>This means the total number of tickets given away must be at least 20.<\/p>\n\n\n\n<p>Next, the total cost of the tickets can be no more than $280. The cost of $10 tickets is ( 10x ), and the cost of $20 tickets is ( 20y ). Therefore, the inequality for the cost of tickets is:<br>[ 10x + 20y \\leq 280 ]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Simplifying the cost inequality<\/h3>\n\n\n\n<p>We can simplify the inequality ( 10x + 20y \\leq 280 ) by dividing the entire inequality by 10:<br>[ x + 2y \\leq 28 ]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Non-negativity constraints<\/h3>\n\n\n\n<p>Additionally, since the number of tickets cannot be negative, we add the non-negativity constraints:<br>[ x \\geq 0 \\quad \\text{and} \\quad y \\geq 0 ]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Graphing the inequalities<\/h3>\n\n\n\n<p>The system of inequalities is:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>( x + y \\geq 20 )<\/li>\n\n\n\n<li>( x + 2y \\leq 28 )<\/li>\n\n\n\n<li>( x \\geq 0 )<\/li>\n\n\n\n<li>( y \\geq 0 )<\/li>\n<\/ol>\n\n\n\n<p>To graph these inequalities:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Graph ( x + y = 20 )<\/strong>: This is a straight line with intercepts at ( (20, 0) ) and ( (0, 20) ). The region above this line satisfies ( x + y \\geq 20 ).<\/li>\n\n\n\n<li><strong>Graph ( x + 2y = 28 )<\/strong>: This is a straight line with intercepts at ( (28, 0) ) and ( (0, 14) ). The region below this line satisfies ( x + 2y \\leq 28 ).<\/li>\n\n\n\n<li>The constraints ( x \\geq 0 ) and ( y \\geq 0 ) limit the solution to the first quadrant.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Identifying the solution region<\/h3>\n\n\n\n<p>The solution region is the area where all inequalities overlap. This region represents the feasible number of $10 and $20 tickets that can be given away while satisfying both the minimum ticket requirement and the cost limitation.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>The solution to this problem is the set of points in the feasible region on the graph. These points represent combinations of $10 and $20 tickets that meet the criteria. To determine the exact number of tickets for specific scenarios, we could test points within this region (like ( x = 10, y = 10 )) to see if they satisfy the inequalities.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A radio station is giving away tickets to a play. They plan to give away tickets for seats that cost $10 and $20. They want to give away at least 20 tickets. The total cost of all the tickets they give away can be no more than $280. Write the inequalities that represent this situation [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-156214","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/156214","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=156214"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/156214\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=156214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=156214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=156214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}