{"id":43,"date":"2017-08-23T07:09:34","date_gmt":"2017-08-23T07:09:34","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=43"},"modified":"2017-09-25T17:02:19","modified_gmt":"2017-09-25T17:02:19","slug":"absolute-values-2","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/absolute-values-2\/","title":{"rendered":"Absolute Values"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/parallel-perpendicular-lines\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/relations-and-functions\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Absolute Values<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study how to solve equations and inequalities containing absolute values.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>In a <em><strong>linear\u00a0equation<\/strong><\/em>, no variable has an exponent greater than\u00a0one.<\/li>\n<li>An <strong><em>inequality <\/em><\/strong>contains one of\u00a0these four symbols:\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p1_clip_image002.gif\" width=\"89\" height=\"20\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<section>\n<h3>What is absolute value?<\/h3>\n<p>The <abbr title=\"the distance of a number from zero on a number line\">absolute\u00a0value<\/abbr> of a number\u00a0<em>x\u00a0<\/em>is its distance from zero on a number line. Absolute value is a\u00a0non-negative number that is concerned only with distance from\u00a0zero, not the direction one must move on the number line to find\u00a0the number. The absolute\u00a0value of <em>x<\/em> is written \u2502<em><span style=\"text-decoration: none;\">x<\/span><\/em>\u2502.<\/p>\n<p>If <em>x\u00a0<\/em>is any real number, then<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"22\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/>= <em>x<\/em> if <em>x<\/em> &gt; 0<\/p>\n<p align=\"center\">\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"22\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/>= 0 if <em>x<\/em> = 0<\/p>\n<p align=\"center\">\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"22\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/> = <em>x<\/em> if <em>x<\/em> &lt; 0<\/p>\n<p>Therefore,<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"22\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/>\u00a0<em>= a<\/em> if and only if <em>x = a <\/em>or<em> x = \u2013a.<\/em><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Remember that the negative of a negative number is a positive. Absolute value is always positive.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the value of <em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em>in the equation \u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"20\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>= 5?<\/p>\n<ol>\n<li>5<\/li>\n<li>\u20135<\/li>\n<li>5 or \u20135<\/li>\n<li>No solution<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C.\u00a0Using the theorem, we see that <em>x\u00a0<\/em>= 5, or <em>x = <\/em>\u20135.\u00a0The solution set is {<em>x<\/em><span style=\"text-decoration: none;\">\u2502\u00a0<em>x = <\/em>5 and <em><span style=\"text-decoration: none;\">x\u00a0= \u2013<\/span><\/em> 5}.<br \/>\n<\/span><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the value of <em>x<\/em> in the equation <em><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p2_html_3ad03e08.gif\" width=\"19\" height=\"22\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>= \u20136?<\/p>\n<ol>\n<li>6<\/li>\n<li>\u20136<\/li>\n<li>6 or \u20136<\/li>\n<li>No solution<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. There is no solution to this problem\u00a0because absolute value cannot be negative.<\/p>\n<\/div>\n<\/section>\n<p>To solve an equation using\u00a0more than one term and absolute value, take what is inside the\u00a0absolute value sign and set it equal to a constant and the\u00a0negative of the constant.<\/p>\n<p>For example, solve the\u00a0equation \u25024<em>x\u00a0+<\/em> 4\u2502 = 44.\u00a0First, set 4<em>x + <\/em>4\u00a0equal to 44 and solve.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p2_clip_image008.gif\" width=\"72\" height=\"59\" name=\"graphics7\" align=\"absmiddle\" border=\"0\" \/><\/center>Then set 4<em><span style=\"text-decoration: none;\">x\u00a0+ <\/span><\/em>4 equal to \u201344\u00a0and solve.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p2_clip_image010.gif\" width=\"87\" height=\"59\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/><\/center>The solution set is {<em>x<\/em>\u2502\u00a0<em>x <\/em>= 10 or\u00a0<em>x <\/em>= \u201312}.<\/p>\n<h3>How do we use absolute value to find the distance between two points on a number line?<\/h3>\n<p>The distance between two values, <em>a<\/em> and\u00a0<em>b<\/em>, is given by\u00a0\u2502<em>b \u2013 a<\/em>\u2502.<\/p>\n<p>Further, if <em>a\u00a0<\/em>and <em>b <\/em><span style=\"text-decoration: none;\">are\u00a0any real numbers, then \u2502<em>a \u2013 b<\/em>\u2502 =\u2502<em>b \u2013 a<\/em>\u2502. <\/span><\/p>\n<p>Again, absolute value is concerned with distance, not\u00a0direction.<\/p>\n<p>If we have two points on a line with coordinates <em>a\u00a0= <\/em>2 and <em>b\u00a0= <\/em>6, then\u00a0the distance between them is<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p3_html_594df044.gif\" width=\"119\" height=\"27\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/><\/center>Because \u2502<em>a \u2013 b<\/em>\u2502 =\u2502<em>b \u2013 a<\/em>\u2502, the distance could also equal <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p3_html_3b17c097.gif\" width=\"159\" height=\"27\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>.<\/p>\n<p>&nbsp;<\/p>\n<p>For example, solve the equation \u25023<em>x\u00a0+ <\/em>5\u2502= 20.<\/p>\n<p>To solve the equation, set what is inside the absolute value\u00a0sign equal to 20 and \u201320 and solve.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p3_clip_image002.gif\" width=\"82\" height=\"221\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/><\/center>The solution set is {<em>x<\/em>\u2502\u00a0<em><span style=\"text-decoration: none;\">x = <\/span><\/em>5 or\u00a0<em>x = \u00a0<u><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p3_clip_image004.gif\" width=\"28\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/u><\/em>}<\/p>\n<h3>How do we solve inequalities that contain absolute\u00a0values?<\/h3>\n<p>When we combine two <abbr title=\"formal statement of inequality between two quantities usually separated by a sign of inequality.\">inequalities<\/abbr> using <em>and<\/em> or <em>or<\/em>,\u00a0we get a <abbr title=\"two or more inequalities taken together\">compound\u00a0inequality<\/abbr>.\u00a0When we combine them using <em>and<\/em>,\u00a0both inequalities must be true. That is, any solutions must solve\u00a0both inequalities. This is called a <abbr title=\"solution to compound inequalities that satisfy all parts. Indicated by \">conjunction<\/abbr>.\u00a0When we combine using <em>or<\/em>,\u00a0a solution is valid if it solves either inequality. That is, a\u00a0valid solution may solve only one inequality. This is called a\u00a0<abbr title=\"solution to compound inequalities that satisfy one part. Indicated by \">disjunction<\/abbr>.<\/p>\n<p style=\"text-decoration: none;\">The following five theorems are\u00a0used for dealing with inequalities containing absolute values.\u00a0Most of these theorems can only be applied for positive numbers.\u00a0However, theorem 1 can also be applied for zero.<\/p>\n<p>For real numbers <em>a &gt; <\/em>0\u00a0and all real numbers <em>x<\/em>,<\/p>\n<p>1. \u2502<em>x<\/em>\u2502 = <em>a<\/em> if and only if <em>x\u00a0= a <\/em>and <em>x\u00a0= \u2013a. <\/em><\/p>\n<p>2. \u2502<em>x<\/em>\u2502\u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image002.gif\" width=\"7\" height=\"8\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/>\u00a0a <\/em>if\u00a0and only if <em>\u2013a <img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image002.gif\" width=\"7\" height=\"8\" name=\"graphics4\" align=\"absmiddle\" border=\"0\" \/>\u00a0x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image003.gif\" width=\"6\" height=\"7\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/>\u00a0a. <\/em><\/p>\n<p>3. \u2502<em>x<\/em>\u2502<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image005.gif\" width=\"9\" height=\"10\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/em> <em>a\u00a0<\/em>if and only if <em>\u2013a\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image005.gif\" width=\"9\" height=\"10\" name=\"graphics7\" align=\"absmiddle\" border=\"0\" \/>\u00a0x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image006.gif\" width=\"9\" height=\"10\" name=\"graphics8\" align=\"absmiddle\" border=\"0\" \/>\u00a0a. <\/em><\/p>\n<p>4. \u2502<em>x<\/em>\u2502\u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image008.gif\" width=\"7\" height=\"8\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em><em><u>a<\/u><\/em> if and only if <em>x\u00a0<\/em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image008.gif\" width=\"7\" height=\"8\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/>\u00a0<span style=\"text-decoration: none;\">a <\/span>or <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image009.gif\" width=\"7\" height=\"8\" name=\"graphics11\" align=\"absmiddle\" border=\"0\" \/> \u2013a. <\/em><\/p>\n<p>5. \u2502<em>x<\/em>\u2502 <em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image011.gif\" width=\"9\" height=\"10\" name=\"graphics12\" align=\"absmiddle\" border=\"0\" \/>\u00a0a <\/em>if and\u00a0only if <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image011.gif\" width=\"9\" height=\"10\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/>\u00a0a <\/em>or <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image012.gif\" width=\"9\" height=\"10\" name=\"graphics14\" align=\"absmiddle\" border=\"0\" \/> \u2013a. <\/em><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the value of <em>x\u00a0<\/em>in the inequality \u2502<em>x<\/em>\u2502\u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image013.gif\" width=\"7\" height=\"8\" name=\"graphics15\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>8<em>?<br \/>\n<\/em><\/p>\n<ol>\n<li>x = 8<\/li>\n<li>\u20138\u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image014.gif\" width=\"7\" height=\"8\" name=\"graphics16\" align=\"absmiddle\" border=\"0\" \/>\u00a0x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image015.gif\" width=\"7\" height=\"8\" name=\"graphics17\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>8<\/li>\n<li>\u20138 <em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image016.gif\" width=\"7\" height=\"8\" name=\"graphics18\" align=\"absmiddle\" border=\"0\" \/>\u00a0x<\/em><\/li>\n<li><em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image017.gif\" width=\"7\" height=\"8\" name=\"graphics19\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>8<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B . To solve, apply theorem 2.<\/p>\n<p align=\"CENTER\"><em>\u20138\u00a0<\/em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image018.gif\" width=\"7\" height=\"8\" name=\"graphics20\" align=\"absmiddle\" border=\"0\" \/>\u00a0x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image019.gif\" width=\"7\" height=\"8\" name=\"graphics21\" align=\"absmiddle\" border=\"0\" \/>\u00a08<\/p>\n<p>The solution set is {<em>x<\/em>\u2502<em>-8\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image020.gif\" width=\"7\" height=\"8\" name=\"graphics22\" align=\"absmiddle\" border=\"0\" \/>\u00a0x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image021.gif\" width=\"7\" height=\"8\" name=\"graphics23\" align=\"absmiddle\" border=\"0\" \/>\u00a08<\/em> } . This\u00a0is a conjunction; to satisfy the inequality, <em>x\u00a0<\/em>must be less than 8 <em>and\u00a0<\/em>greater than \u20138.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the value of <em>x\u00a0<\/em><span style=\"text-decoration: none;\">in the inequality \u2502<em>x<\/em>\u2502<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image022.gif\" width=\"9\" height=\"10\" name=\"graphics24\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>15?<br \/>\n<\/span><\/p>\n<ol>\n<li><em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image023.gif\" width=\"9\" height=\"10\" name=\"graphics25\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>\u201315<\/li>\n<li><em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image024.gif\" width=\"9\" height=\"10\" name=\"graphics26\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>15<\/li>\n<li><em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image025.gif\" width=\"9\" height=\"10\" name=\"graphics27\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em><span style=\"text-decoration: none;\">15 and <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image026.gif\" width=\"9\" height=\"10\" name=\"graphics28\" align=\"absmiddle\" border=\"0\" \/><\/em> \u201315<br \/>\n<\/span><\/li>\n<li><em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image027.gif\" width=\"9\" height=\"10\" name=\"graphics29\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>15 or <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image028.gif\" width=\"9\" height=\"10\" name=\"graphics30\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>\u201315<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. To solve, apply theorem 5. Set <em>x\u00a0<\/em>greater than or equal to 15 or less than or equal to \u201315.\u00a0This forms the disjunction <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image029.gif\" width=\"9\" height=\"10\" name=\"graphics31\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>15 or <em>x\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image030.gif\" width=\"9\" height=\"10\" name=\"graphics32\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>\u201315.<\/p>\n<\/div>\n<\/section>\n<p>In one last example, solve\u00a054 <em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image031.gif\" width=\"9\" height=\"10\" name=\"graphics33\" align=\"absmiddle\" border=\"0\" \/><\/em><span style=\"text-decoration: none;\"> \u25026<em>x \u2013\u00a0<\/em>18\u2502 for <em>x<\/em><em>.<\/em><br \/>\n<\/span><\/p>\n<p>To solve, apply theorem 3.\u00a0Form the conjunction <em>\u2013<\/em>54\u00a0<em><img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image032.gif\" width=\"9\" height=\"10\" name=\"graphics34\" align=\"absmiddle\" border=\"0\" \/>\u00a0<\/em>6<em>x \u2013 <\/em>18\u00a0<img loading=\"lazy\" decoding=\"async\" class=\" no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s2_p4_clip_image033.gif\" width=\"9\" height=\"10\" name=\"graphics35\" align=\"absmiddle\" border=\"0\" \/>\u00a054<em>, <\/em>and\u00a0then solve.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p4_html_1cd536c5.gif\" width=\"92\" height=\"187\" name=\"graphics36\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>The solution is <em><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s2_p4_html_m479c0658.gif\" width=\"77\" height=\"19\" name=\"graphics37\" align=\"absmiddle\" border=\"0\" \/>.<\/em><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/parallel-perpendicular-lines\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/relations-and-functions\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Absolute Values Objective In this lesson, you will study how to solve equations and inequalities containing absolute values. Previously Covered: In a linear\u00a0equation, no variable has an exponent greater than\u00a0one. An inequality contains one of\u00a0these four symbols:\u00a0 What is absolute value? The absolute\u00a0value of a number\u00a0x\u00a0is its distance from [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-43","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/43","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=43"}],"version-history":[{"count":10,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/43\/revisions"}],"predecessor-version":[{"id":833,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/43\/revisions\/833"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}