[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267254-105":59,"doc-detail-267254-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","absolute-value-equations-and-inequalities-linear-and-non-linear","Absolute Value Equations and Inequalities - Linear and Non-Linear","","This guide explains how to solve absolute value equations and inequalities by isolating the absolute value, translating it into equivalent cases based on whether it equals zero, a positive value, or a negative value, and then solving the resulting linear systems. It covers solution logic for both “one solution” and “two solutions” scenarios, plus interval notation for inequalities. Non-linear cases require substitution back into the original equation to eliminate extraneous solutions. Includes worked examples and practice problems.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/absolute-value-equations-and-inequalities-linear-and-non-linear/267254/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/absolute-value-equations-and-inequalities-linear-and-non-linear/267254.png","ImageObject",300,407,{"name":92,"@type":93},"Jiven","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",21,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you start solving a linear absolute value equation like |A| = k?","Question",{"text":113,"@type":114},"Isolate the absolute value first. Then analyze the value it equals: if k = 0, remove the absolute value and solve for one solution; if k is negative, there is no solution; if k is positive, set the argument equal to k and to -k and solve both.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"When do linear absolute value inequalities have one solution, no solution, or all real numbers?",{"text":118,"@type":114},"The guide uses case logic based on the inequality relative to zero: if the absolute value is less than zero, there is no solution; if it is less than or equal to zero, there is one solution (set the argument equal to zero); if it is greater than or equal to zero, the solution is all real numbers.",{"name":120,"@type":111,"acceptedAnswer":121},"Why are non-linear absolute value solutions sometimes extraneous?",{"text":122,"@type":114},"Because solving introduces candidate answers that must be validated. The guide requires plugging each candidate back into the original non-linear equation to check validity and discard extraneous solutions that do not satisfy the original equation.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},267254,1789416359,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","Absolute Value Equations and Inequalities  \nAbsolute Value Definition - The absolute value of x, is defined as…  \n|􀀃 | = 􀀅 􀀆, , 􀀈􀀉􀀈􀀉􀀆􀀆 00 where x is called the “argument”  \nSteps for Solving Linear Absolute Value Equations: i.e. |􀀎􀀆 + 􀀐| = 􀀑  \n1. Isolate the absolute value.  \n2. Identify what the isolated absolute value is set equal to…  \na. If the absolute value is set equal to zero, remove absolute value symbols & solve the equation to get one solution.  \nb. If the absolute value is set equal to a negative number, there is no solution.  \nc. If the absolute value is set equal to a positive number, set the argument (expression within the absolute value) equal to the number and set it equal to the opposite of the number, using an ‘or’ statement in between the two equations. Then solve each equation separately to get two solutions.  \nExamples:  \na. |3􀀆 + 12| + 7 = 7  \n|3􀀆 + 12| = 0  \nBecause this equals  \n0, there is ONE solution.  \n3􀀆 + 12 = 0  \nb. |3􀀆 − 7| + 7 = 2  \n|3􀀆 − 7| = −5  \nBecause this equalsa negative number, there is NO solution.  \nNo Solution  \nc.  \n3􀀆 − 7 = 2  \n|3􀀆 − 7| + 7 = 9  \n|3􀀆 − 7| = 2  \nBecause this equalsa positive number there are TWO sltns.  \nor 3􀀆 − 7 = −2  \n3􀀆 = −12 3􀀆 = 9 or 3􀀆 = 5  \n􀀃 = −􀀚 􀀃 = 􀀛 or 􀀃 = 􀀜􀀛  \nd. |􀀆 + 5| = |2􀀆 − 1| 􀀁  \n􀀆 + 5 = +(2􀀆 − 1)  \n􀀃 = 􀀟  \nSet up two Equations  \nor 􀀆 + 5 = −(2􀀆 − 1)  \nor 􀀆 + 5 = −2􀀆 + 1 → 3􀀆 = −4 → 􀀃 = -  \n􀀚  \n􀀛  \nSteps for Solving Linear Absolute Value Inequalities: 􀀈 . 􀀣 |􀀎􀀆 + 􀀐| ≤ 􀀑  \n1. Isolate the absolute value.  \n2. Identify what the absolute value inequality is set “equal” to…  \na. If the absolute value is less than zero, there is no solution.  \nb. If the absolute value is less than or equal to zero, there is one solution. Just set the argument equal to zero and solve.  \nc. If the absolute value is greater than or equal to zero, the solution is all real numbers.  \nd. If the absolute value is greater than zero, the solution is all real numbers except for the value which makes it equal to zero. This will be written as a union.  \ne. If the absolute value is less than or less than or equal to a negative number, there is no solution. The absolute value of something will never be less than or equal to a negative number.  \nf. If the absolute value is greater than or greater than or equal to a negative number, the solution is all real numbers. The absolute value of something will always be greater than a negative number.  \ng. If the absolute value is less than or less than or equal to a positive number, the problem can be approached two ways. Either way, the solution will be written as an intersection.  \ni. Place the argument in a 3-part inequality (compound) between the opposite of the number and the number, then solve.  \nii. Set the argument less than the number and greater than the opposite of the number using an “and” statement in between the two inequalities.  \nh. If the absolute value is greater than or greater than or equal to a positive number, set the argument less than the opposite of the number and greater than the number using an ‘or’ statement in between the two inequalities. Then solve each inequality, writing the solution as a union of the two solutions.  \n3. Graph the answer on a number line and write the answer in interval notation.  \nExamples:  \na. |􀀆 − 4| ≥ 0 b. |2􀀆 − 1| + 4 \u003C 4 c. −3 + |􀀆 + 1| ≤ −3  \n|2􀀆 − 1| \u003C 0 |􀀆 + 1| ≤ 0  \nAll Real Numbers No Solution Set 􀀆 + 1 = 0  \nSo 􀀃 = −􀀥  \nd. |3􀀆 + 4| + 5 ≤ 3 |3􀀆 + 4| ≤ −2  \nNo Solution  \ne. 2|􀀆 − 1| − 4 ≥ 2 f. 2|􀀆 − 1| ≥ 6  \n|􀀆 − 1| ≥ 3  \n􀀆 − 1 ≥ 3 􀀧􀀨 􀀆 − 1 ≤ −3  \n􀀆 ≥ 4 􀀧􀀨 􀀆 ≤ −2 (−∞, −􀀪􀀫 ∪ 􀀭􀀚, ∞)  \n|􀀆 − 6| + 6 ≥ −4 |􀀆 − 6| ≥ −10  \nAll Real Numbers  \ng. |2 − 􀀆| \u003C 8  \n2 − 􀀆 \u003C 8 􀀧􀀨 2 − 􀀆 > −8  \n−􀀆 \u003C −6 􀀧􀀨 − 􀀆 > −10 􀀆 > 6 􀀧􀀨 􀀆 \u003C 10  \n(􀀟, 􀀥􀀱)  \nh. 3|4􀀆 − 1| ≤ 9 |4􀀆 − 1| ≤ 3  \ni. |􀀆 + 6| > 0 Set 􀀆 + 6 ≠ 0 So 􀀆 ≠ −6  \n(−∞, −􀀟) ∪ (−􀀟, ∞)  \nOption 1 – Split in to two different Inequalities joined by an “AND”statement (Intersection)  \n3|4􀀆 − 1| ≤ 9  \n|4􀀆 − 1| ≤ 3  \n4􀀆 − 1 ≤ 3 􀀲􀀳􀀴 4􀀆 − 1 ≥ −3  \n􀀆 ≤ 1 􀀲􀀳􀀴 􀀆 ≥","cbCaiqDteKIZOp3Z","https://ap.wps.com/l/cbCaiqDteKIZOp3Z","pdf",119911,"English","# Absolute Value Definition\n# Steps for Solving Linear Absolute Value Equations\n## Isolate the absolute value\n## Case analysis: zero, negative, positive\n## Examples\n# Steps for Solving Linear Absolute Value Inequalities\n## Isolate the absolute value\n## Case analysis with unions and intersections\n## Graphing and interval notation\n## Examples\n# Steps for Solving NON-Linear Absolute Value Equations\n## Check answers and remove extraneous solutions\n## Examples\n# Absolute Value Practice Problems","[{\"question\":\"How do you start solving a linear absolute value equation like |A| = k?\",\"answer\":\"Isolate the absolute value first. Then analyze the value it equals: if k = 0, remove the absolute value and solve for one solution; if k is negative, there is no solution; if k is positive, set the argument equal to k and to -k and solve both.\"},{\"question\":\"When do linear absolute value inequalities have one solution, no solution, or all real numbers?\",\"answer\":\"The guide uses case logic based on the inequality relative to zero: if the absolute value is less than zero, there is no solution; if it is less than or equal to zero, there is one solution (set the argument equal to zero); if it is greater than or equal to zero, the solution is all real numbers.\"},{\"question\":\"Why are non-linear absolute value solutions sometimes extraneous?\",\"answer\":\"Because solving introduces candidate answers that must be validated. The guide requires plugging each candidate back into the original non-linear equation to check validity and discard extraneous solutions that do not satisfy the original equation.\"}]","Absolute Value Equations and Inequalities - Linear and Non-Linear | PDF",15]