{"id":688,"date":"2013-05-28T14:33:43","date_gmt":"2013-05-28T14:33:43","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=688"},"modified":"2026-04-27T13:51:20","modified_gmt":"2026-04-27T18:51:20","slug":"linear-inequalities","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/linear-inequalities\/","title":{"rendered":"Graphing Linear Inequalities"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_KSr0ABSwD-E\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_KSr0ABSwD-E\" data-source-videoID=\"KSr0ABSwD-E\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Graphing Linear Inequalities Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Graphing Linear Inequalities\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_KSr0ABSwD-E:hover {cursor:pointer;} img#videoThumbnailImage_KSr0ABSwD-E {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1130-linear-inequalities-resized-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_KSr0ABSwD-E\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_KSr0ABSwD-E\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Graphing Linear Inequalities\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_KSr0ABSwD-E\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_KSr0ABSwD-E\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_KSr0ABSwD-E\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction lke_Function() {\n  var x = document.getElementById(\"lke\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"lke_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"lke\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_an_Inequality\" class=\"smooth-scroll\">What is an Inequality?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_with_Two_Variables\" class=\"smooth-scroll\">Graphing with Two Variables<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Graphing_Linear_Inequalities_Practice_Questions\" class=\"smooth-scroll\">Graphing Linear Inequalities Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"overview\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"overview\">Overview<\/label><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"overview-spoiler\">\n<h2><span id=\"What_is_an_Inequality\" class=\"m-toc-anchor\"><\/span>What is an Inequality?<\/h2>\n<p>An <a class=\"ylist\" href=\"..\/inequalities\/\">inequality<\/a> is a mathematical expression where the two sides are not equal.<\/p>\n<p>There are 4 inequality symbols:<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 40%;\">\n<tbody>\n<tr>\n<td>\\(\\gt\\)<\/td>\n<td>Greater than<\/td>\n<\/tr>\n<tr>\n<td>\\(\\lt\\)<\/td>\n<td>Less than<\/td>\n<\/tr>\n<tr>\n<td>\\(\\geq\\)<\/td>\n<td>Greater than or equal to<\/td>\n<\/tr>\n<tr>\n<td>\\(\\leq\\)<\/td>\n<td>Less than or equal to<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"margin-top: 1em;\">We read the inequality, \\(y\u22642x+6\\), as \u201c\\(y\\) is less than or equal to six more than two times \\(x\\)\u201d.<\/p>\n<h2><span id=\"Graphing_with_Two_Variables\" class=\"m-toc-anchor\"><\/span>Graphing with Two Variables<\/h2>\n<p>To graph a linear inequality with two variables, we use the same process as when <a class=\"ylist\" href=\"..\/graphing-linear-equations\/\">graphing a linear equation<\/a> plus a few extra steps. When graphing the line, we either make our line solid, which includes the values on the line as part of the solution set, or we use a dotted line, which does not include the values on the line as part of the solution set. Here is how we decide if the line should be solid or dotted:<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 35%;\">\n<tbody>\n<tr>\n<td>\\(\\gt\\)<\/td>\n<td>Dotted line<\/td>\n<\/tr>\n<tr>\n<td>\\(\\lt\\)<\/td>\n<td>Dotted line<\/td>\n<\/tr>\n<tr>\n<td>\\(\\geq\\)<\/td>\n<td>Solid line<\/td>\n<\/tr>\n<tr>\n<td>\\(\\leq\\)<\/td>\n<td>Solid line<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"margin-top: 1em;\">The last step in the process is to decide which side of the line to shade. One method is to pick one point on each side of the line, substitute into the equation, and simplify. Whichever results in a true statement will be the side that is shaded.<\/p>\n<p>Another method is to shade below the line if the inequality symbol is \\(\\lt\\) (less than) or \\(\\leq\\) (less than or equal to), and shade above the line if the inequality symbol is \\(\\gt\\) (greater than) or \\(\\geq\\) (greater than or equal to).<\/p>\n<h3><span id=\"Example_Problem\" class=\"m-toc-anchor\"><\/span>Example Problem<\/h3>\n<p>Graph the inequality \\(y\u22642x+6\\).<\/p>\n<p>Start by graphing the line as if it is an equation, \\(y=2x+6\\). Since it is already in <a class=\"ylist\" href=\"..\/slope-intercept-and-point-slope-forms\/\">slope-intercept form<\/a>, we know that the \\(y\\)-intercept is \\(6\\) and the slope is \\(2\\). Since the inequality is \\(\\leq\\), we will make the line solid.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-105708 aligncenter\" src=\"..\/wp-content\/uploads\/2021\/12\/Graphing-Two-Variable-Linear-Inequalities-O-SQ-1.png\" alt=\"Linear line on a graph\" width=\"603\" height=\"595\" \/>As a last step, we pick one point on each side of the line and plug it into the inequality. For ease, I am going to pick \\((0,0)\\), on the right side and \\((-4,4)\\) on the left side.<\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<tbody>\n<tr>\n<td>\\((0,0)\\)<\/td>\n<td>\\((-4,4)\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(0\\leq 2(0)+6\\)<\/td>\n<td>\\(4\\leq 2(-4)+6\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(0\\leq 6\\)<\/p>\n<p style=\"text-align: center; margin-bottom: 0em;\">True statement<\/p>\n<\/td>\n<td>\\(4\\leq -2\\)<\/p>\n<p style=\"text-align: center; margin-bottom: 0em;\">False statement<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"margin-top: 1em;\">We will shade the side that gave us the true statement because this means that the ordered pair is part of the solution set of the inequality.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-105711 aligncenter\" src=\"..\/wp-content\/uploads\/2021\/12\/Graphing-Two-Variable-Linear-Inequalities-O-SQ-2.png\" alt=\"Linear line graphed with the right side shaded\" width=\"603\" height=\"595\" \/><\/p>\n<\/div>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about linear inequalities! Today, we\u2019ll explore what linear inequalities are, and see how to express their solutions.<\/p>\n<h3><span id=\"Understanding_Linear_Inequalities\" class=\"m-toc-anchor\"><\/span>Understanding Linear Inequalities<\/h3>\n<p>\nYou probably have some experience with linear equations. They can be written in point-slope form \\(y-y_{1}=m(x-x_{1})\\), slope-intercept form \\((y=mx + b)\\), or standard form \\((Ax + By = C)\\) and their graphs are straight lines.<\/p>\n<p>For an example, let\u2019s use the equation \\(y=2x + 1\\).<\/p>\n<p>There are an infinite number of solutions to the equation. Solutions, in this case, are coordinate pairs that lie on the line\u2014algebraically, they make the equation true. So \\((0,1)\\) is a solution because \\(1 = 2(0) + 1\\). \\((-1,-1)\\) is also a solution since \\(-1 = 2(-1) + 1\\). <\/p>\n<p>On the other hand, \\((3,2)\\) is not a solution because \\(2 \u2260 2(3) + 1\\).<\/p>\n<p>What if, instead of the equation \\(y = 2x + 1\\), we replaced the equals sign with \\(\\lt\\) to get \\(y > 2x + 1\\)?  Well, for starters, that would give us a linear inequality! <\/p>\n<p>Linear inequalities are simply inequalities that involve linear equations.<\/p>\n<h3><span id=\"Testing_Points_on_a_Linear_Inequality\" class=\"m-toc-anchor\"><\/span>Testing Points on a Linear Inequality<\/h3>\n<p>\nThe solution set is still an infinite set of points, but let\u2019s see where they lie. We\u2019ll start by graphing the line \\(y = 2x + 1\\) as before.<\/p>\n<p>Okay, now let\u2019s make a quick table so we can test some points on this graph. We\u2019re interested in points on each side of the line, as well as on the line. <\/p>\n<p>Let\u2019s test \\((3,-1)\\) first. We can see this is below the line, and if we plug these numbers into our inequality, we see that it does not work. Or, we would say it isn\u2019t \u201ctrue\u201d.<\/p>\n<p>Let\u2019s test another point, \\((2,1)\\). This is also below the line, and plugging the values into our inequality tells us it is not true.<\/p>\n<p>Let\u2019s try some that are on the line. \\((1,3)\\) and \\((0,1)\\) are both points on the line, but plugging in those values still doesn\u2019t work.<\/p>\n<p>Now let\u2019s try above the line. \\((-3,3)\\) is above the line, and if we plug those values into our inequality, we find that it is true:  \\(2(-3) + 1\\). \\((-1,0)\\) is also above the line, and if we plug those values into our inequality, we find that it is true as well: \\(2(-1) + 1\\).<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 70%;\">\n<thead>\n<tr>\n<th>Location<\/th>\n<th>Point<\/th>\n<th>True?<\/th>\n<\/thead>\n<tbody>\n<tr>\n<td>Below line<\/td>\n<td>\\((3,-1)\\)<\/td>\n<td>\\(-1 \\gt 2(3) + 1\\)?\u00a0 <strong>No.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Below line<\/td>\n<td>\\((2,1)\\)<\/td>\n<td>\\(1 \\gt 2(2) + 1\\)?\u00a0 <strong>No.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>On line<\/td>\n<td>\\((1,3)\\)<\/td>\n<td>\\(3 \\gt 2(1) + 1\\)?\u00a0 <strong>No.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>On line<\/td>\n<td>\\((0,1)\\)<\/td>\n<td>\\(1 \\gt 2(0) + 1\\)? <strong>No.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Above line<\/td>\n<td>\\((-3,3)\\)<\/td>\n<td>\\(3 \\gt 2(-3) + 1\\)? <strong>Yes.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Above line<\/td>\n<td>\\((-1,0)\\)<\/td>\n<td>\\(0 \\gt 2(-1) + 1\\)? <strong>Yes.<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nWe can test as many points as we\u2019d like, but this tells us more than enough information.  All the points in the solution set are above the line, not on it. To show that, we shade that entire half of the graph.<\/p>\n<p>The reason all of the points in the solution set are above the line is because of our inequality symbol. Our solution is \u201cgreater than\u201d, so it will be above the line. If we used \\(\\lt\\), our solution would be below the line.<\/p>\n<p>So, that\u2019s really the whole concept of linear inequalities. <\/p>\n<h3><span id=\"Steps_to_Solve_and_Graph_Linear_Inequalities\" class=\"m-toc-anchor\"><\/span>Steps to Solve and Graph Linear Inequalities<\/h3>\n<p>\nIn practice, we\u2019re talking about 3 key steps:<\/p>\n<ol>\n<li>Graph the line as normal.<\/li>\n<li>Test 1 or 2 points; \\((0,0)\\) is always the easiest. If it lies on the line, try something like \\((1,1)\\) or \\((-1,1)\\).<\/li>\n<li>Dash the line if needed. If the inequality contains a less than or greater than, use a dashed line. If it contains \\(\\leq\\) or \\(\\geq\\), keep it solid.<\/li>\n<\/ol>\n<p>Let\u2019s use these steps for the inequality \\(3x + 6y \u2264 -12\\)!<\/p>\n<p>Here\u2019s the graph.<\/p>\n<p>Let\u2019s test \\((0,0)\\): \\(3(0) + 6(0) \u2264 -12\\) is not true. Therefore, we shade in the direction that does not include \\((0,0)\\) and keep the line solid.<\/p>\n<p>Before we go, let\u2019s have a little practice!<\/p>\n<p>Write the inequality represented by this graph. Pause the video if you need more time:<\/p>\n<p>The correct answer is \\(y \u2265 4x &#8211; 8\\).<\/p>\n<p>I hope this review was helpful! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Graphing_Linear_Inequalities_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Graphing Linear Inequalities Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich graph best represents the inequality \\(y>-2x+4\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Solving-linear-inequalities-graph-1.svg\" alt=\"A graph shows the inequality y \u2265 x + 2. The region above the dashed line y = x + 2 is shaded. The axes are labeled from -5 to 5.\" width=\"400.95\" height=\"401.5\" class=\"alignnone size-full wp-image-275269\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-2\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Solving-linear-inequalities-graph-2.svg\" alt=\"A coordinate plane shows the shaded region below the dashed line y = -x + 4, representing the solution to the inequality y &lt; -x + 4.\" width=\"400.95\" height=\"401.5\" class=\"alignnone size-full wp-image-275272\"  role=\"img\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Solving-linear-inequalities-graph-3.svg\" alt=\"A graph with a dashed line representing the equation x + 2y = 4, shading the region above and to the right of the line in blue.\" width=\"400.95\" height=\"400.95\" class=\"alignnone size-full wp-image-275260\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-4\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Solving-linear-inequalities-graph-4.svg\" alt=\"Cartesian graph showing the line y = -x + 4. The region above the line, including the boundary, is shaded blue. The x- and y-axes range from -5 to 5.\" width=\"400.95\" height=\"400.95\" class=\"alignnone size-full wp-image-275263\"  role=\"img\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, graph the line for \\(y>-2x+4\\) on the coordinate plane. Since the \\(y\\)-intercept (\\(b\\)) is 4, the line will intersect the \\(y\\)-axis at 4, or \\((0,4)\\). The slope (\\(m\\)) is -2 or \\(-\\frac{2}{1}\\). Starting at \\((0,4)\\), use the slope to find another point on the line. Since slope is rise over run, move down two units and right one unit to the point \\((1,2)\\). <\/p>\n<p>Next, draw a line to connect the points. Since the inequality sign is \u201cgreater than\u201d, connect the ordered pairs with a dashed line. This indicates that points on this line are not possible solutions. <\/p>\n<p>Then, substitute origin point \\((0,0)\\) into the inequality as a test point to determine which portion of the coordinate plane to shade:<\/p>\n<p style=\"text-align: center; line-height: 35px;\">\\(y>-2x+4\\)<br \/>\n\\((0)>-2(0)+4\\)<br \/>\n\\(0>0+4\\)<br \/>\n\\(0>4\\) (incorrect)<\/p>\n<p>Since 0 is not greater than 4, the point \\((0,0)\\) cannot be part of the solution set. That means the values below the graphed line are not solutions to the inequality. Instead, shade the area above the line. The finished graph has a dashed line with a shaded region above it. All possible solutions are points above the line graphed.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich ordered pair shows a possible solution for the inequality \\(y\u2264\\frac{1}{2}x-5\\)? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">\\((5,-5)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\((0,0)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\((5,5)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\((-5,5)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Substitute each point into the inequality as a test point to determine if it is part of the solution set.<\/p>\n<p style=\"text-align: center; line-height: 45px;\">\\(y \\leq \\frac{1}{2}x-5\\)<br \/>\n\\((-5) \\leq \\frac{1}{2}(5)-5\\)<br \/>\n\\(-5 \\leq 2\\frac{1}{2}-5\\)<br \/>\n\\(-5 \\leq -2\\frac{1}{2}\\)<\/p>\n<p>Since \u22125 is less than or equal to \\(-2\\frac{1}{2}\\), the point \\((5,-5)\\) is part of the solution set.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich inequality statement matches the graph shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Solving-linear-inequalities-graph-5.svg\" alt=\"A dashed line shows the equation y = x, with the region above and to the right of the line shaded blue on a Cartesian grid.\" width=\"397.1\" height=\"396.55\" class=\"aligncenter size-full wp-image-275266\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\(y\u22643x+2\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">\\(y\\) < \\(3x+2\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(y\\) < \\(3x-2\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(y\u2264-3x+2\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>First, identify the \\(y\\)-intercept by looking at where the graph of the line crosses the \\(y\\)-axis. The \\(y\\)-intercept (\\(b\\)) is 2, or \\((0,2)\\).<\/p>\n<p>Since C has \u22122 as the \\(y\\)-intercept (\\(b\\)), we can eliminate this answer choice.<\/p>\n<p>Because the graph moves in a positive direction, the slope (\\(m\\)) is a positive number. Therefore, we can eliminate D as an answer choice, which has \u22123 for the slope (\\(m\\)).<\/p>\n<p>The inequality is graphed with a dashed line, which indicates that points on the line are not possible solutions. In other words, the inequality symbol must be either \\(\\lt\\) or \\(\\gt\\). We can eliminate A because the inequality contains the symbol \\(\\leq\\), which is graphed as a solid line.<\/p>\n<p>Therefore, the correct answer is B.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nVanessa is at a used bookstore that sells pre-owned books and DVDs. DVDs cost $8 each, and books cost $5 each. Vanessa wants to buy as many books and DVDs as she can afford, but she can\u2019t spend more than $35. Which inequality statement best represents this situation? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">\\(8x+5y \\geq 35\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">\\(8x+5y \\leq 35\\) <\/div><div class=\"PQ\"  id=\"PQ-4-3\">\\(8x+5y \\gt 35\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">\\(8x+5y \\lt 35\\) <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The inequality statement needs to show that the total amount of money Vanessa spends cannot exceed $35. In other words, Vanessa can spend less than $35 or exactly $35. The inequality sign that matches this scenario is \u201cless than or equal to\u201d (\\(\\leq\\)). <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMatthew is at the movie theater with a group of friends. He plans to buy some bags of popcorn and sodas for the group. Each bag of popcorn costs $6, and each soda costs $4. Matthew has $40 to spend and writes the inequality \\(6x+4y\u226440\\) to represent the situation.<\/p>\n<p>Which statement below is true? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Matthew can buy 6 bags of popcorn and 2 sodas for his friends. <\/div><div class=\"PQ\"  id=\"PQ-5-2\">Matthew can buy 3 bags of popcorn and 6 sodas for his friends. <\/div><div class=\"PQ\"  id=\"PQ-5-3\">Matthew can buy 5 bags of popcorn and 3 sodas for his friends. <\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">Matthew can buy 4 bags of popcorn and 4 sodas for his friends. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Using \\(x\\) to represent bags of popcorn and \\(y\\) to represent sodas, substitute each set of values into the inequality as a test point. Doing so will determine if it is part of the solution set.<\/p>\n<p style=\"text-align: center; line-height: 35px;\">\n\\(6x+4y\u226440\\)<br \/>\n\\(6(4)+4(4)\u226440\\)<br \/>\n\\(24+16\u226440\\)<br \/>\n\\(40\u226440\\)<\/p>\n<p>Since 40 is less than <em>or equal to<\/em> 40, the set of values in Choice D are part of the solution set. The values given for Choices A, B, and C are not correct because they do not yield true statements when substituted into the inequality. Therefore, D is the correct answer.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #6:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich shows the graph of the inequality \\(y\u2264-\\frac{2}{3}x+4\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-6-1\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Linear-Inequalities-Example-A.svg\" alt=\"A graph of the line y = -x + 3 on an x-y axis, with the region below the line shaded. The axes range from -6 to 6.\" width=\"473.6\" height=\"477.6\" class=\"alignnone size-full wp-image-287645\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-6-2\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Linear-Inequalities-Example-B.svg\" alt=\"A graph with a shaded region above the line y = -x + 6, including the line, on a coordinate plane with x and y axes labeled.\" width=\"473.6\" height=\"477.6\" class=\"alignnone size-full wp-image-287648\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-6-3\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Linear-Inequalities-Example-C.svg\" alt=\"A line with positive slope passes through the origin on an x-y coordinate grid, moving upward from left to right. The grid axes are labeled from -6 to 6.\" width=\"473.6\" height=\"477.6\" class=\"alignnone size-full wp-image-287651\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-6-4\"><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Linear-Inequalities-Example-D.svg\" alt=\"A graph with the line y = x, shading the region below and to the right of the line, on a coordinate plane with labeled axes.\" width=\"473.6\" height=\"477.6\" class=\"alignnone size-full wp-image-287654\"  role=\"img\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-6\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-6\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-6-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To graph an inequality, we start by graphing it as a linear equation.<\/p>\n<p>Since \\(y \\leq -\\frac{2}{3}x+4\\) is in slope-intercept form, we know the \\(y\\)-intercept is \\(4\\) and the slope is \\(-\\frac{2}{3}\\).<\/p>\n<p>Once the line is graphed, we will determine if the line should be solid or dotted. In this case since it is \\(\\leq\\) (less than or equal to), the line will be solid, which means the values on the line are part of the solution set for the inequality.<\/p>\n<p>The last step is to determine which side of the line should be shaded. One method is to pick a point one and use it to evaluate the inequality. If the point that results in a true statement, it will be on the side that is shaded.<\/p>\n<p>Start with \\((0,0)\\), \\(0\\leq -\\frac{2}{3}(0)+4\\), after simplifying we are left with \\(0 \\leq 4\\), which is a true statement, so we will shade below the line.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-6-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-6-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #7:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nHere is the graph of an inequality:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Linear-Inequalities-Example-Q7.svg\" alt=\"A graph shows the line y = x + 2 with shading above the line, indicating the region y \u2265 x + 2. The x- and y-axes range from -6 to 6.\" width=\"473.6\" height=\"477.6\" class=\"aligncenter size-full wp-image-287657\"  role=\"img\" \/><\/p>\n<p>Which inequality is shown on the graph?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-7-1\">\\(y\u2265\\frac{1}{2}x+2\\)<\/div><div class=\"PQ\"  id=\"PQ-7-2\">\\(y\u2264\\frac{1}{2}x+2\\)<\/div><div class=\"PQ\"  id=\"PQ-7-3\">\\(y\u22652x+\\frac{1}{2}\\)<\/div><div class=\"PQ\"  id=\"PQ-7-4\">\\(y\u22642x-\\frac{1}{2}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-7\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-7\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-7-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph of the line shows a line that is going through \\((0,2)\\) and slope of \\(\\frac{1}{2}\\), since the top part of the graph is shaded, and the line is solid, the correct answer is \\(y\u2265\\frac{1}{2}x+2\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-7-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-7-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #8:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of an inequality is shown.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/04\/Inequality-graph.svg\" alt=\"A coordinate plane shows the line y = x + 2 with a dashed boundary and the region below the line shaded.\" width=\"477\" height=\"477\" class=\"aligncenter size-full wp-image-292523\"  role=\"img\" \/><\/p>\n<p>Which inequality is shown on the graph?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-8-1\">\\(y \\leq -\\frac{5}{3}x-5\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-8-2\">\\(y \\lt -\\frac{5}{3}x-5\\)<\/div><div class=\"PQ\"  id=\"PQ-8-3\">\\(y \\geq -5x-\\frac{5}{3}\\)<\/div><div class=\"PQ\"  id=\"PQ-8-4\">\\(y \\gt -5x-\\frac{5}{3}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-8\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-8\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-8-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The dotted line, which tells us that the answer will either have a \\(\\gt\\) or \\(\\lt\\) inequality symbol, goes through \\((0,-5)\\), which is the \\(y\\)-intercept, and has a slope of \\(-\\frac{5}{3}\\).<\/p>\n<p>Since the bottom half of the graph is shaded, we know the answer is \\(y \\lt -\\frac{5}{3}x-5\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-8-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-8-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #9:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of an inequality has a dotted line, is shaded on top, has a slope of \u22122 and goes through the point \\((0,3)\\). Which inequality has this graph?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-9-1\">\\(y \\lt -2x+3\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-9-2\">\\(y \\gt -2x+3\\)<\/div><div class=\"PQ\"  id=\"PQ-9-3\">\\(y\\leq-2x+3\\)<\/div><div class=\"PQ\"  id=\"PQ-9-4\">\\(y\\geq-2x+3\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-9\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-9\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-9-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since all the inequalities have a slope of \u22122 and their \\(y\\)-intercept is \\((0,3)\\), we will take a closer look at the inequality symbols.<\/p>\n<p>The inequality that would produce the graph described would have to have either a \\(\\lt\\) or \\(\\gt\\) symbol since it is a dotted line. The fact that the top part of the graph, above the line, is shaded tells us that it must be the \\(\\gt\\) symbol. Therefore, correct answer is \\(y \\gt -2x+3\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-9-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-9-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #10:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of an inequality goes through the points \\((0,-6)\\) and \\((3,0)\\), it is a solid line and is shaded on the bottom half of the line. Which inequality has this graph?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-10-1\">\\(y \\lt -6x+2\\)<\/div><div class=\"PQ\"  id=\"PQ-10-2\">\\(y \\gt -6x+2\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-10-3\">\\(y \\leq 2x-6\\)<\/div><div class=\"PQ\"  id=\"PQ-10-4\">\\(y\\geq2x-6\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-10\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-10\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-10-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When given two points, we can use the slope formula to find the slope of the line, which is the change in \\(y\\) over the change in \\(x\\):<\/p>\n<p style=\"text-align: center\">\\(m=\\dfrac{0-(-6)}{3-0}=2\\)<\/p>\n<p>We are told that the graph of the line goes through the point \\((0,-6)\\), so this is the \\(y\\)-intercept. Given that the \\(y\\)-intercept is \u22126, the slope is 2, and the line is solid with the bottom half shaded, the correct answer is \\(y \\leq 2x-6\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-10-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-10-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":100246,"parent":0,"menu_order":36,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-688","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-inequalities-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Linear inequalities are graphed with a dotted or solid line and then shading based on the sign. 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