{"id":382,"date":"2025-09-25T22:49:55","date_gmt":"2025-09-25T22:49:55","guid":{"rendered":"https:\/\/ssstt.app\/news\/?p=382"},"modified":"2025-09-25T22:50:04","modified_gmt":"2025-09-25T22:50:04","slug":"how-to-solve-compound-inequalities","status":"publish","type":"post","link":"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/","title":{"rendered":"How to solve compound inequalities"},"content":{"rendered":"<p class=\"break-words\" dir=\"auto\">Solving compound inequalities involves finding the values of a variable that satisfy multiple inequalities simultaneously. Compound inequalities are typically connected by &#8220;and&#8221; (conjunction, where both conditions must be true) or &#8220;or&#8221; (disjunction, where at least one condition must be true). Here\u2019s a clear, step-by-step guide to solve them, with examples for both types.<\/p>\n<ul>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>How to solve compound inequalities step by step<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>Compound inequality calculator<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>How to solve compound inequalities pdf<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>How to solve inequalities<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>Compound inequality examples with solutions<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>Compound inequalities examples<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>How to solve compound inequalities with fractions<\/strong><\/mark><\/a><br \/>\n<mark><\/mark><\/li>\n<li dir=\"auto\"><a href=\"https:\/\/ssstt.app\/news\/how-to-solve-compound-inequalities\/\"><mark><strong>How to graph compound inequalities<\/strong><\/mark><\/a><\/li>\n<\/ul>\n<h3 class=\"text-xl\" dir=\"auto\">General Steps for Solving Compound Inequalities<\/h3>\n<p><iframe loading=\"lazy\" title=\"Solving Compound Inequalities\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/aFXo2ws-roU?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<ol class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">Identify the Type of Compound Inequality<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">&#8220;And&#8221; Inequality<\/strong>: Both conditions must be satisfied (e.g., <span class=\"katex\"><span class=\"katex-mathml\">\u22123&lt;x&lt;5 -3 &lt; x &lt; 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">3<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>).<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">&#8220;Or&#8221; Inequality<\/strong>: At least one condition must be satisfied (e.g., <span class=\"katex\"><span class=\"katex-mathml\">x&lt;\u22122 x &lt; -2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">x&gt;3 x &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>).<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Break It Down<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">For &#8220;and&#8221; inequalities, solve each part separately or treat them as a single expression if written compactly (e.g., <span class=\"katex\"><span class=\"katex-mathml\">a&lt;x&lt;b a &lt; x &lt; b <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>).<\/li>\n<li class=\"break-words\">For &#8220;or&#8221; inequalities, solve each inequality independently.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Solve Each Inequality<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Use standard inequality-solving techniques (similar to solving equations):\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Isolate the variable by adding, subtracting, multiplying, or dividing both sides.<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Reverse the inequality sign<\/strong> if multiplying or dividing by a negative number.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\">Simplify to find the solution for each part.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Combine the Solutions<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">For &#8220;and&#8221;: Find the intersection of the solutions (where both conditions overlap).<\/li>\n<li class=\"break-words\">For &#8220;or&#8221;: Find the union of the solutions (all values that satisfy at least one condition).<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Graph the Solution (Optional)<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Plot the solution on a number line to visualize the range of values.<\/li>\n<li class=\"break-words\">Use closed dots (<span class=\"katex\"><span class=\"katex-mathml\">\u2219\\bullet<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2219<\/span><\/span><\/span><\/span>) for <span class=\"katex\"><span class=\"katex-mathml\">\u2264\\leq<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">\u2264<\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">\u2265\\geq<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">\u2265<\/span><\/span><\/span><\/span>, and open dots (<span class=\"katex\"><span class=\"katex-mathml\">\u2218\\circ<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2218<\/span><\/span><\/span><\/span>) for <span class=\"katex\"><span class=\"katex-mathml\">&lt;&lt;<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">&lt;<\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">&gt;&gt;<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">&gt;<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Shade the appropriate regions (overlapping for &#8220;and,&#8221; separate for &#8220;or&#8221;).<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Write the Solution<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Express in interval notation or inequality notation, depending on the requirement.<\/li>\n<li class=\"break-words\">Check for special cases (e.g., no solution or all real numbers).<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"text-xl\" dir=\"auto\">Example 1: &#8220;And&#8221; Compound Inequality<\/h3>\n<p class=\"break-words\" dir=\"auto\">Solve: <span class=\"katex\"><span class=\"katex-mathml\">\u22122&lt;3x+1&lt;10 -2 &lt; 3x + 1 &lt; 10 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span><\/p>\n<ol class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">Treat as a Single Expression<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Since it\u2019s an &#8220;and&#8221; inequality, solve the entire expression at once.<\/li>\n<li class=\"break-words\">Start with: <span class=\"katex\"><span class=\"katex-mathml\">\u22122&lt;3x+1&lt;10 -2 &lt; 3x + 1 &lt; 10 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Isolate the Variable<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Subtract 1 from all parts: <span class=\"katex\"><span class=\"katex-mathml\">\u22122\u22121&lt;3x+1\u22121&lt;10\u22121 -2 &#8211; 1 &lt; 3x + 1 &#8211; 1 &lt; 10 &#8211; 1 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\">\u22123&lt;3x&lt;9 -3 &lt; 3x &lt; 9 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">3<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">9<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Divide all parts by 3: <span class=\"katex\"><span class=\"katex-mathml\">\u221233&lt;3&#215;3&lt;93 \\frac{-3}{3} &lt; \\frac{3x}{3} &lt; \\frac{9}{3} <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u22123<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">9<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\">\u22121&lt;x&lt;3 -1 &lt; x &lt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Solution<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Inequality notation: <span class=\"katex\"><span class=\"katex-mathml\">\u22121&lt;x&lt;3 -1 &lt; x &lt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Interval notation: <span class=\"katex\"><span class=\"katex-mathml\">(\u22121,3) (-1, 3) <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Graph: On a number line, shade between -1 and 3 with open dots at -1 and 3 (since <span class=\"katex\"><span class=\"katex-mathml\">&lt;&lt;<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">&lt;<\/span><\/span><\/span><\/span>).<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Check<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Test <span class=\"katex\"><span class=\"katex-mathml\">x=0 x = 0 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\">\u22122&lt;3(0)+1&lt;10\u2192\u22122&lt;1&lt;10 -2 &lt; 3(0) + 1 &lt; 10 \\rightarrow -2 &lt; 1 &lt; 10 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span>, which is true.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"text-xl\" dir=\"auto\">Example 2: &#8220;Or&#8221; Compound Inequality<\/h3>\n<p class=\"break-words\" dir=\"auto\">Solve: <span class=\"katex\"><span class=\"katex-mathml\">2x\u22123\u22645 2x &#8211; 3 \\leq 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">x+1&gt;4 x + 1 &gt; 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><\/p>\n<ol class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">Solve Each Inequality Separately<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">First inequality: <span class=\"katex\"><span class=\"katex-mathml\">2x\u22123\u22645 2x &#8211; 3 \\leq 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Add 3: <span class=\"katex\"><span class=\"katex-mathml\">2x\u22648 2x \\leq 8 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Divide by 2: <span class=\"katex\"><span class=\"katex-mathml\">x\u22644 x \\leq 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\">Second inequality: <span class=\"katex\"><span class=\"katex-mathml\">x+1&gt;4 x + 1 &gt; 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Subtract 1: <span class=\"katex\"><span class=\"katex-mathml\">x&gt;3 x &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Combine Solutions<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">For &#8220;or,&#8221; take the union: <span class=\"katex\"><span class=\"katex-mathml\">x\u22644 x \\leq 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">x&gt;3 x &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">On a number line, this includes all values <span class=\"katex\"><span class=\"katex-mathml\">x\u22644 x \\leq 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> (from negative infinity to 4) and all values <span class=\"katex\"><span class=\"katex-mathml\">x&gt;3 x &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span> (from 3 to positive infinity).<\/li>\n<li class=\"break-words\">Since the regions overlap at <span class=\"katex\"><span class=\"katex-mathml\">x&gt;3 x &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>, the solution is all real numbers.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Solution<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Inequality notation: All real numbers.<\/li>\n<li class=\"break-words\">Interval notation: <span class=\"katex\"><span class=\"katex-mathml\">(\u2212\u221e,\u221e) (-\\infty, \\infty) <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">\u221e<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">\u221e<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Graph: The entire number line is shaded.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Check<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Test <span class=\"katex\"><span class=\"katex-mathml\">x=0 x = 0 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\">2(0)\u22123=\u22123\u22645 2(0) &#8211; 3 = -3 \\leq 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">3<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> (true), so it satisfies the first part.<\/li>\n<li class=\"break-words\">Test <span class=\"katex\"><span class=\"katex-mathml\">x=5 x = 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\">5+1=6&gt;4 5 + 1 = 6 &gt; 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> (true), so it satisfies the second part.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"text-xl\" dir=\"auto\">Example 3: &#8220;And&#8221; Inequality Written Separately<\/h3>\n<p class=\"break-words\" dir=\"auto\">Solve: <span class=\"katex\"><span class=\"katex-mathml\">2x+1&gt;3 2x + 1 &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">x\u22124\u22642 x &#8211; 4 \\leq 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/p>\n<ol class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">Solve Each Inequality<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">First: <span class=\"katex\"><span class=\"katex-mathml\">2x+1&gt;3 2x + 1 &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Subtract 1: <span class=\"katex\"><span class=\"katex-mathml\">2x&gt;2 2x &gt; 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Divide by 2: <span class=\"katex\"><span class=\"katex-mathml\">x&gt;1 x &gt; 1 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\">Second: <span class=\"katex\"><span class=\"katex-mathml\">x\u22124\u22642 x &#8211; 4 \\leq 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Add 4: <span class=\"katex\"><span class=\"katex-mathml\">x\u22646 x \\leq 6 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Combine Solutions<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">For &#8220;and,&#8221; take the intersection: <span class=\"katex\"><span class=\"katex-mathml\">x&gt;1 x &gt; 1 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">x\u22646 x \\leq 6 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">This means <span class=\"katex\"><span class=\"katex-mathml\">1&lt;x\u22646 1 &lt; x \\leq 6 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Solution<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Inequality notation: <span class=\"katex\"><span class=\"katex-mathml\">1&lt;x\u22646 1 &lt; x \\leq 6 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Interval notation: <span class=\"katex\"><span class=\"katex-mathml\">(1,6] (1, 6] <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">6<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">Graph: Shade between 1 and 6, with an open dot at 1 and a closed dot at 6.<\/li>\n<\/ul>\n<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Check<\/strong>:\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Test <span class=\"katex\"><span class=\"katex-mathml\">x=2 x = 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\">2(2)+1=5&gt;3 2(2) + 1 = 5 &gt; 3 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span> (true) and <span class=\"katex\"><span class=\"katex-mathml\">2\u22124=\u22122\u22642 2 &#8211; 4 = -2 \\leq 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span> (true).<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"text-xl\" dir=\"auto\">Special Cases<\/h3>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\"><strong class=\"font-semibold\">No Solution<\/strong>: If the conditions can\u2019t overlap (e.g., <span class=\"katex\"><span class=\"katex-mathml\">x&lt;2 x &lt; 2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">x&gt;5 x &gt; 5 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>), the solution is empty (<span class=\"katex\"><span class=\"katex-mathml\">\u2205\\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>).<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">All Real Numbers<\/strong>: If the solution covers all numbers (as in Example 2), use <span class=\"katex\"><span class=\"katex-mathml\">(\u2212\u221e,\u221e) (-\\infty, \\infty) <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">\u221e<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">\u221e<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\"><strong class=\"font-semibold\">Negative Coefficients<\/strong>: If you multiply\/divide by a negative number, flip the inequality sign. Example: Solve <span class=\"katex\"><span class=\"katex-mathml\">\u22122x&gt;4 -2x &gt; 4 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span>. Divide by -2: <span class=\"katex\"><span class=\"katex-mathml\">x&lt;\u22122 x &lt; -2 <\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<h3 class=\"text-xl\" dir=\"auto\">Tips<\/h3>\n<ul class=\"marker:text-secondary\" dir=\"auto\">\n<li class=\"break-words\">Always double-check by testing a value in the solution range.<\/li>\n<li class=\"break-words\">If graphing, ensure the correct use of open\/closed dots based on <span class=\"katex\"><span class=\"katex-mathml\">&lt;&lt;<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">&lt;<\/span><\/span><\/span><\/span> vs. <span class=\"katex\"><span class=\"katex-mathml\">\u2264\\leq<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">\u2264<\/span><\/span><\/span><\/span>.<\/li>\n<li class=\"break-words\">For complex inequalities, write each step clearly to avoid mistakes.<\/li>\n<\/ul>\n<p class=\"break-words\" dir=\"auto\">If you have a specific compound inequality to solve or want a visual graph (I can provide a description or guide you to plot it), let me know!<\/p>\n<blockquote>\n<ol class=\"slider-wrapper\">\n<li id=\"slide-565\" class=\"cue-slide\">\n<div>\n<p><strong>Example 1:<\/strong>\u00a0Solve the compound inequality -7 &lt; -3x &#8211; 2 \u2264 5 and represent the solution in the interval notation.<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>The given inequality is,<\/p>\n<p>-7 &lt; -3x &#8211; 2 \u2264 5<\/p>\n<p>Adding 2 on all the sides,<\/p>\n<p>-5 &lt; -3x \u2264 7<\/p>\n<p>Dividing all the sides by -3 (note that the signs of inequalities change as we are dividing by negative number),<\/p>\n<p>5\/3 &gt; x \u2265 -7\/3<\/p>\n<p>This can be written as -7\/3 \u2264 x &lt; 5\/3.<\/p>\n<p>Hence, the solution in the interval notation is [-7\/3, 5\/3).<\/p>\n<p><strong>Answer:<\/strong>\u00a0[-7\/3, 5\/3)<\/p>\n<\/div>\n<p>&nbsp;<\/li>\n<\/ol>\n<\/blockquote>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-large wp-image-383\" src=\"https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100-614x1024.jpg\" alt=\"How to solve compound inequalities\" width=\"614\" height=\"1024\" srcset=\"https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100-614x1024.jpg 614w, https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100-180x300.jpg 180w, https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100-768x1280.jpg 768w, https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100-512x853.jpg 512w, https:\/\/ssstt.app\/news\/wp-content\/uploads\/2025\/09\/How-to-solve-compound-inequalities-100.jpg 900w\" sizes=\"auto, (max-width: 614px) 100vw, 614px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"Solving compound inequalities involves finding the values of a variable that satisfy multiple inequalities simultaneously. Compound inequalities are&hellip;","protected":false},"author":1,"featured_media":383,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"csco_display_header_overlay":false,"csco_singular_sidebar":"","csco_page_header_type":"","footnotes":""},"categories":[3],"tags":[],"class_list":{"0":"post-382","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-trends","8":"cs-entry"},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/posts\/382","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/comments?post=382"}],"version-history":[{"count":3,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/posts\/382\/revisions"}],"predecessor-version":[{"id":387,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/posts\/382\/revisions\/387"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/media\/383"}],"wp:attachment":[{"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/media?parent=382"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/categories?post=382"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ssstt.app\/news\/wp-json\/wp\/v2\/tags?post=382"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}