{"id":222759,"date":"2025-05-31T17:37:06","date_gmt":"2025-05-31T17:37:06","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222759"},"modified":"2025-05-31T17:37:08","modified_gmt":"2025-05-31T17:37:08","slug":"penelope-went-to-the-store-to-buy-x-pounds-of-broccoli-for-1-10-per-pound-and-y-cans-of-soup-for-2-50-each-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/31\/penelope-went-to-the-store-to-buy-x-pounds-of-broccoli-for-1-10-per-pound-and-y-cans-of-soup-for-2-50-each-2\/","title":{"rendered":"Penelope went to the store to buy x pounds of broccoli for $1.10 per pound and y cans of soup for $2.50 each"},"content":{"rendered":"\n<p>Penelope went to the store to buy x pounds of broccoli for $1.10 per pound and y cans of soup for $2.50 each. In total, she spent less than $10. The inequality relating the purchases she made and the total purchase price is 1.10x + 2.50y &lt; 10. Which are reasonable solutions for this situation? Check all that apply. (\u20131, 4) (0, 2) (3, 2.5) (2, 4) (0.5, 3.78) (1.5, 3)<br>Penelope went to the store to buy x pounds of broccoli for<br>2.50 each. In total, she spent less than $10. The inequality relating the purchases she made and the total purchase price is 1.10x + 2.50y &lt; 10. Which are reasonable solutions for this situation? Check all that apply. (\u20131, 4) (0, 2) (3, 2.5) (2, 4) (0.5, 3.78) (1.5, 3)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this problem, we are given the inequality: 1.10x+2.50y&lt;101.10x + 2.50y &lt; 10<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>xx = pounds of broccoli bought at $1.10 per pound<\/li>\n\n\n\n<li>yy = cans of soup bought at $2.50 each<\/li>\n\n\n\n<li>The total spent must be <strong>less than $10<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>We are given several coordinate pairs (x, y) to test as possible solutions. For a solution to be <strong>reasonable<\/strong>, it must:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Satisfy the inequality 1.10x+2.50y&lt;101.10x + 2.50y &lt; 10<\/li>\n\n\n\n<li>Make sense in a real-world context, where xx and yy must be <strong>non-negative<\/strong> (you can\u2019t buy negative quantities)<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Test Each Option:<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">(\u20131, 4)<\/h4>\n\n\n\n<p>1.10(\u22121)+2.50(4)=\u22121.10+10=8.90&lt;10\u21d2Satisfies&nbsp;inequality&nbsp;but&nbsp;x=\u22121&nbsp;is&nbsp;not&nbsp;valid1.10(-1) + 2.50(4) = -1.10 + 10 = 8.90 &lt; 10 \\Rightarrow \\text{Satisfies inequality but } x = -1 \\text{ is not valid}<\/p>\n\n\n\n<p><strong>Reject<\/strong> (Negative broccoli is not possible)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(0, 2)<\/h4>\n\n\n\n<p>1.10(0)+2.50(2)=0+5=5&lt;101.10(0) + 2.50(2) = 0 + 5 = 5 &lt; 10<\/p>\n\n\n\n<p><strong>Accept<\/strong> (Reasonable and satisfies the inequality)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(3, 2.5)<\/h4>\n\n\n\n<p>1.10(3)+2.50(2.5)=3.30+6.25=9.55&lt;101.10(3) + 2.50(2.5) = 3.30 + 6.25 = 9.55 &lt; 10<\/p>\n\n\n\n<p><strong>Accept<\/strong> (Reasonable even with fractional soup cans, though rare in real life, still mathematically valid)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(2, 4)<\/h4>\n\n\n\n<p>1.10(2)+2.50(4)=2.20+10=12.20&gt;101.10(2) + 2.50(4) = 2.20 + 10 = 12.20 &gt; 10<\/p>\n\n\n\n<p><strong>Reject<\/strong> (Exceeds $10 budget)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(0.5, 3.78)<\/h4>\n\n\n\n<p>1.10(0.5)+2.50(3.78)=0.55+9.45=10.00&lt;\u0338101.10(0.5) + 2.50(3.78) = 0.55 + 9.45 = 10.00 \\not&lt; 10<\/p>\n\n\n\n<p><strong>Reject<\/strong> (Equals $10, but not <em>less than<\/em> $10)<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">(1.5, 3)<\/h4>\n\n\n\n<p>1.10(1.5)+2.50(3)=1.65+7.50=9.15&lt;101.10(1.5) + 2.50(3) = 1.65 + 7.50 = 9.15 &lt; 10<\/p>\n\n\n\n<p><strong>Accept<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(0, 2)<\/strong><\/li>\n\n\n\n<li><strong>(3, 2.5)<\/strong><\/li>\n\n\n\n<li><strong>(1.5, 3)<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation :<\/h3>\n\n\n\n<p>This problem involves identifying valid real-world solutions to a linear inequality modeling a shopping budget. Penelope is buying xx pounds of broccoli at $1.10 per pound and yy cans of soup at $2.50 each. The constraint is that her total purchase must be <strong>less than $10<\/strong>, which gives the inequality: 1.10x+2.50y&lt;101.10x + 2.50y &lt; 10<\/p>\n\n\n\n<p>To determine which points are valid, each given pair (x, y) is substituted into the inequality. Only those combinations that result in a value <strong>strictly less than 10<\/strong> qualify.<\/p>\n\n\n\n<p>Additionally, because this is a real-life situation involving buying physical goods, values of xx and yy must be <strong>non-negative<\/strong>. Negative quantities like \u20131 pounds of broccoli aren&#8217;t realistic, so such points are discarded even if they mathematically satisfy the inequality.<\/p>\n\n\n\n<p>Evaluating each point:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(\u20131, 4) satisfies the inequality mathematically but is not realistic.<\/li>\n\n\n\n<li>(0, 2), (3, 2.5), and (1.5, 3) are all reasonable and satisfy the budget constraint.<\/li>\n\n\n\n<li>(2, 4) exceeds the $10 limit.<\/li>\n\n\n\n<li>(0.5, 3.78) hits exactly $10, which doesn&#8217;t meet the &#8220;less than $10&#8221; requirement.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the <strong>reasonable solutions<\/strong> that satisfy both the inequality and the context of the situation are:<br><strong>(0, 2), (3, 2.5), and (1.5, 3)<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner9-97.jpeg\" alt=\"\" class=\"wp-image-222760\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Penelope went to the store to buy x pounds of broccoli for $1.10 per pound and y cans of soup for $2.50 each. In total, she spent less than $10. The inequality relating the purchases she made and the total purchase price is 1.10x + 2.50y &lt; 10. Which are reasonable solutions for 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-222759","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222759","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=222759"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222759\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222759"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222759"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222759"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}