Absolute Value - GMAT Math Study Guide

Another means to understand absolute value is as the distance from the origin. For example, at the value of x = -7, the distance from the origin, where x = 0, is the absolute value of x=-7, which is 7.

Solving Equations With Absolute Values

An equation with an absolute value term within it often has two solutions. In order to solve an absolute value equation: (1) isolate the absolute value term (2) Solve the absolute value term for two circumstances: (i) the value inside the absolute value brackets is positive (ii) the value within the absolute value brackets is negative.

To summarize: 1.) Isolate the absolute value term. 2.) Set the expression opposite the absolute value as negative (i.e., multiply it by negative one).

Another Example:

Another Example:

Working With Multiple Absolute Values

Multiple absolute values should be dealt with by working with the innermost one and moving outwards. For example:

Types of GMAT Problems

1. Solving Equations With Absolute Values

   Begin by isolating the absolute value. Rewrite the new equation without the absolute value brackets and solve for the variable. Rewrite the new equation without the absolute value brackets and multiply one side of the equation by -1. Solve this new equation for the variable. The solution should consist of two different answers, usually one being negative and the other being positive (although this is not always true).

   Solve for x, |x+5|-10 = 0

   A)	x=5 or x=-15
   B)	x=2 or x=-18
   C)	x=3 or x=-17
   D)	x=6 or x=-14
   E)	x=1 or x=-17

   Correct Answer: A

   1. Isolate the absolute value.
      |x+5|-10 = 0
      |x+5| = 10; (add 10 to both sides)
   2. x+5 can equal +10 or -10 due to absolute value
   3. Rewrite the equation without the brackets.
      x+5 = 10
   4. Solve for x by subtracting 5 from both sides.
      x = 10 - 5
      x=5
   5. Rewrite |x+5| = 10 without the brackets and multiply the right side by -1 since the expression in the absolute value brackets could be negative due to the effect of the absolute value.
      x+5 = (10)(-1)
      x+5 = -10
   6. Solve for x by subtracting 5 from both sides.
      x = -10 - 5
      x=-15
   7. x = 5 or x = -15
2. Solving Inequalities With Absolute Values

   Begin by isolating the absolute value. Rewrite the new equation without the absolute value brackets and solve for the variable. Rewrite the new equation without the absolute value brackets, flip the inequality, and multiply the side that did not have the absolute value by -1. Solve this new equation for the variable. The solution will correspond to a section (or sections) of the number line.

   Solve for x, |3x-5|-5 < -2x

   A)	x>1 and x<-3
   B)	0<x<2
   C)	-2<x<2
   D)	x>2 and x<-5
   E)	x>1 and x<0

   Correct Answer: B

   1. Rewrite the inequality so that the absolute value is isolated.
      |3x-5|-5 < -2x
      |3x-5| < -2x+5
   2. Remove the brackets.
      3x-5 < -2x+5
   3. Solve for x.
      3x+2x < 5+5 [added 2x to both sides; added 5 to both sides]
      5x < 10
      x < 2
   4. Rewrite |3x-5| < -2x+5 without the brackets and multiply the right side by -1.
      Since you are multiplying an inequality by a negative number, you must flip the inequality sign.
      3x-5 > (-2x+5)(-1)
      3x-5 > 2x-5
      
   5. Solve for x.
      3x-2x > -5+5
      x > 0
      
   6. Combine the two solutions to get: 0<x<2

© 2015 Lighthouse Prep, LLC. GMAT™ is a registered trademark of The Graduate Management Admission Council™ (GMAC), which does not endorse nor is affiliated in any way with the owner of this website or any content contained herein. All content (including practice questions and study guides) is written by PlatinumPrep, LLC not GMAC.