GMAT Math will ask you about absolute values.\u00a0 Mastering what the GMAT asks about them requires sophisticated understanding.<\/p>\n
Somewhere along the line, perhaps in middle school, you probably learned:<\/p>\n
|positive| = positive and |negative| = positive<\/p>\n
<\/p>\n
In other words, the equation |x| = 5 has the solution: x = 5 or<\/strong> x = \u22125.\u00a0 (Notice: the word \"or\" is not a garnish there; it's actually an essential piece of mathematical equipment.)<\/p>\n <\/p>\n That's great, but the GMAT is simply not going to ask you to solve the equation |x| = 5.\u00a0 When the GMAT asks about absolute value, it's going to be something more in the vein of |3x \u2013 7| = 5.\u00a0 The basic idea is (as is often the case in more advanced algebra) is to replace \"x\" in the simpler equation above with whatever \"thing\" is between the absolute value.\u00a0 If q is a positive constant, then<\/p>\n |thing| = q<\/p>\n has the solution:<\/p>\n thing = q or<\/strong> thing = -q<\/p>\n In the given example,<\/p>\n |3x \u2013 7| = 5<\/p>\n 3x \u2013 7 = 5\u00a0or <\/strong>3x \u2013 7 = -5<\/p>\n 3x = 12 or<\/strong> 3x = 2<\/p>\n x = 4 or<\/strong> x = 3\/2<\/p>\n That's an example of an absolute value equation<\/span>, which the GMAT could ask.\u00a0 The GMAT is even more likely to ask about an absolute value inequality.<\/p>\n <\/p>\n OK, let's face it.\u00a0 The definition of absolute value that says \"keeps a positive positive, and makes a negative positive\" \u2013 the utility of that definition peaks in middle school.\u00a0 We need to have a more sophisticated understanding of absolute value to handle everything the GMAT will ask of it.<\/p>\n Here is the more sophisticated definition of absolute value.\u00a0 The absolute value of x, |x| is the distance<\/span> of x from zero on the number line<\/strong>.\u00a0 Of course, it's always positive, because distance is away positive.<\/p>\n To extend that further: |x \u2013 4| is the distance of x from 4; |x \u2013 7| is the distance of x from 7;|x + 3| is the distance of x from -3 (this is because x + 3 = x \u2013 (-3) when written as subtraction).<\/p>\n That is profoundly important in solving the absolute value inequalities that the GMAT will ask of you.\u00a0 Suppose a GMAT Math question asks you: represent the region -1 < x < 9 as an absolute value inequality.<\/p>\n The first step is to find the midpoint of the region: 4 is exactly halfway between -1 and 9.\u00a0 Now the distances: 9 is a distance of 5 from 4, and so is -1.\u00a0 So the distance from 4 (viz. |x \u2013 4|) can't equal 5, but it can be anything up to 5.\u00a0 Thus<\/p>\n |x \u2013 4|< 5<\/p>\n is the absolute value inequality representation of the region -1 < x < 9.\u00a0 Integrate this understanding, and you will be able to handle anything the GMAT asks you about absolute value.<\/p>\n <\/p>\n Here's a free Magoosh practice problem, a challenging absolute value practice question<\/a>, with a video explanation of the answer.<\/p>\n <\/p>\nExpanding the Pattern<\/h2>\n
Rethinking Absolute Value<\/h2>\n