{"id":224820,"date":"2023-10-03T09:11:11","date_gmt":"2023-10-03T09:11:11","guid":{"rendered":"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/"},"modified":"2023-10-03T09:11:11","modified_gmt":"2023-10-03T09:11:11","slug":"what-is-the-solution-for-the-following-absolute-value-inequality","status":"publish","type":"post","link":"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/","title":{"rendered":"What is the solution for the following absolute value inequality?"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_62 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title \" >Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#What_is_the_solution_for_the_following_absolute_value_inequality\" title=\"What is the solution for the following absolute value inequality?\">What is the solution for the following absolute value inequality?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#FAQs\" title=\"FAQs:\">FAQs:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#1_What_is_an_absolute_value_inequality\" title=\"1. What is an absolute value inequality?\">1. What is an absolute value inequality?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#2_How_do_you_solve_an_absolute_value_inequality_with_a_positive_expression_inside\" title=\"2. How do you solve an absolute value inequality with a positive expression inside?\">2. How do you solve an absolute value inequality with a positive expression inside?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#3_How_do_you_solve_an_absolute_value_inequality_with_a_negative_expression_inside\" title=\"3. How do you solve an absolute value inequality with a negative expression inside?\">3. How do you solve an absolute value inequality with a negative expression inside?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#4_Can_absolute_value_inequalities_have_more_than_one_solution\" title=\"4. Can absolute value inequalities have more than one solution?\">4. Can absolute value inequalities have more than one solution?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#5_What_happens_if_the_inequality_symbol_is_strict_such_as_%3C_or_%3E\" title=\"5. What happens if the inequality symbol is strict, such as &lt; or &gt;?\">5. What happens if the inequality symbol is strict, such as < or >?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#6_Are_the_solutions_to_absolute_value_inequalities_always_real_numbers\" title=\"6. Are the solutions to absolute value inequalities always real numbers?\">6. Are the solutions to absolute value inequalities always real numbers?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#7_Can_absolute_value_inequalities_be_solved_using_graphical_methods\" title=\"7. Can absolute value inequalities be solved using graphical methods?\">7. Can absolute value inequalities be solved using graphical methods?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#8_Are_there_any_cases_where_absolute_value_inequalities_have_no_solution\" title=\"8. Are there any cases where absolute value inequalities have no solution?\">8. Are there any cases where absolute value inequalities have no solution?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#9_Can_the_solution_to_an_absolute_value_inequality_be_an_empty_set\" title=\"9. Can the solution to an absolute value inequality be an empty set?\">9. Can the solution to an absolute value inequality be an empty set?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#10_Can_the_solution_to_an_absolute_value_inequality_be_an_infinite_set\" title=\"10. Can the solution to an absolute value inequality be an infinite set?\">10. Can the solution to an absolute value inequality be an infinite set?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#11_What_if_the_absolute_value_inequality_contains_multiple_absolute_value_expressions\" title=\"11. What if the absolute value inequality contains multiple absolute value expressions?\">11. What if the absolute value inequality contains multiple absolute value expressions?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#12_Are_there_any_shortcuts_or_tricks_to_solve_absolute_value_inequalities_more_easily\" title=\"12. Are there any shortcuts or tricks to solve absolute value inequalities more easily?\">12. Are there any shortcuts or tricks to solve absolute value inequalities more easily?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_solution_for_the_following_absolute_value_inequality\"><\/span>What is the solution for the following absolute value inequality?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Absolute value inequalities can be solved by considering two cases: when the value inside the absolute value symbol is positive and when it is negative. Let&#8217;s solve an example to understand the process.<\/p>\n<p>Consider the inequality: |3x &#8211; 4| \u2264 7<\/p>\n<p>To solve this, we need to divide it into two cases:<\/p>\n<p>Case 1: (3x &#8211; 4) \u2265 0<br \/>\nWhen the expression inside the absolute value is non-negative, we can remove the absolute value symbols without changing the inequality sign. Therefore, in this case, we have:<br \/>\n3x &#8211; 4 \u2264 7<\/p>\n<p>Solving this linear inequality:<br \/>\n3x \u2264 11<br \/>\nx \u2264 11\/3<\/p>\n<p>Case 2: (3x &#8211; 4) < 0<br \/>\nWhen the expression inside the absolute value is negative, we need to multiply the inequality sign by -1 and flip the inequality direction. Thus, in this case, we have:<br \/>\n-(3x &#8211; 4) \u2264 7<\/p>\n<p>Simplifying:<br \/>\n-3x + 4 \u2264 7<br \/>\n-3x \u2264 3<br \/>\nx \u2265 -1<\/p>\n<p>Combining the solutions from both cases:<br \/>\nThe solution for the absolute value inequality |3x &#8211; 4| \u2264 7 is -1 \u2264 x \u2264 11\/3.<\/p>\n<hr>\n<p><\/p>\n<h3><span class=\"ez-toc-section\" id=\"FAQs\"><\/span>FAQs:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1_What_is_an_absolute_value_inequality\"><\/span>1. What is an absolute value inequality?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nAn absolute value inequality involves an absolute value expression (such as |x|) and a relational inequality symbol (such as \u2264 or \u2265).<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2_How_do_you_solve_an_absolute_value_inequality_with_a_positive_expression_inside\"><\/span>2. How do you solve an absolute value inequality with a positive expression inside?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nWhen the expression inside the absolute value is non-negative, you can remove the absolute value symbols without changing the inequality sign.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3_How_do_you_solve_an_absolute_value_inequality_with_a_negative_expression_inside\"><\/span>3. How do you solve an absolute value inequality with a negative expression inside?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nWhen the expression inside the absolute value is negative, multiply the inequality sign by -1 and flip the inequality direction.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4_Can_absolute_value_inequalities_have_more_than_one_solution\"><\/span>4. Can absolute value inequalities have more than one solution?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, absolute value inequalities can have multiple solutions. The solution to an absolute value inequality often consists of a range of values.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5_What_happens_if_the_inequality_symbol_is_strict_such_as_%3C_or_%3E\"><\/span>5. What happens if the inequality symbol is strict, such as < or >?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nIf the inequality symbol is strict, such as < or >, the solution will not include the boundary points. They will be excluded from the solution.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6_Are_the_solutions_to_absolute_value_inequalities_always_real_numbers\"><\/span>6. Are the solutions to absolute value inequalities always real numbers?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, the solutions to absolute value inequalities are always real numbers, unless the inequality is impossible to satisfy.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7_Can_absolute_value_inequalities_be_solved_using_graphical_methods\"><\/span>7. Can absolute value inequalities be solved using graphical methods?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, absolute value inequalities can indeed be solved graphically. The graph of the absolute value expression can help determine the solution range.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"8_Are_there_any_cases_where_absolute_value_inequalities_have_no_solution\"><\/span>8. Are there any cases where absolute value inequalities have no solution?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, there are cases where absolute value inequalities have no solution, such as when the absolute value expression is less than zero and the inequality symbol is \u2265 or \u2264.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"9_Can_the_solution_to_an_absolute_value_inequality_be_an_empty_set\"><\/span>9. Can the solution to an absolute value inequality be an empty set?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nYes, it is possible for the solution to an absolute value inequality to be an empty set if the inequality cannot be satisfied under any circumstances.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"10_Can_the_solution_to_an_absolute_value_inequality_be_an_infinite_set\"><\/span>10. Can the solution to an absolute value inequality be an infinite set?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nNo, the solution to an absolute value inequality is always a finite set or an empty set. It cannot be an infinite set.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"11_What_if_the_absolute_value_inequality_contains_multiple_absolute_value_expressions\"><\/span>11. What if the absolute value inequality contains multiple absolute value expressions?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nIf an absolute value inequality contains multiple absolute value expressions, each expression should be treated separately, considering the conditions for both positive and negative values.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"12_Are_there_any_shortcuts_or_tricks_to_solve_absolute_value_inequalities_more_easily\"><\/span>12. Are there any shortcuts or tricks to solve absolute value inequalities more easily?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\nNo specific shortcuts or tricks apply to all absolute value inequalities, as each one may have different conditions and expressions. It is important to carefully analyze and solve each inequality on its own merits.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is the solution for the following absolute value inequality? Absolute value inequalities can be solved by considering two cases: when the value inside the absolute value symbol is positive and when it is negative. Let&#8217;s solve an example to understand the process. Consider the inequality: |3x &#8211; 4| \u2264 7 To solve this, we &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"What is the solution for the following absolute value inequality?\" class=\"read-more button\" href=\"https:\/\/namso-gen.co\/blog\/what-is-the-solution-for-the-following-absolute-value-inequality\/#more-224820\">Read more<span class=\"screen-reader-text\">What is the solution for the following absolute value inequality?<\/span><\/a><\/p>\n","protected":false},"author":56,"featured_media":107420,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[86279],"tags":[],"class_list":["post-224820","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-learn","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is the solution for the following absolute value inequality?<\/title>\n<meta name=\"description\" content=\"What is the solution for the following absolute value inequality? 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