The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

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WORKSHEET 3.16: SOLVING ABSOLUTE VALUE INEQUALITIES

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The following guidelines will help you to rewrite and solve absolute value inequalities:

• Isolate the absolute value expression.


• When the expression within the absolute value symbols is<or≤a positive number,
  the inequality is equivalent to a conjunction. The absolute value expression is<or≤
  the positive numberand>or≥the opposite of the positive number.


• When the expression within the absolute value symbols is>or≥a positive number,
  the inequality is equivalent to a disjunction. The absolute value expression is>or≥
  the positive numberor<or≤the opposite of the positive number.


• After rewriting the absolute value inequality as a conjunction or disjunction, the con-
  junction or disjunction can be solved.

EXAMPLES

|x+ 3 |< 6 | 2 x|≥ 10

conjunction disjunction

x+ 3 <6andx+ 3 >− 62 x≥10 or 2x≤− 10

x<3andx>− 9 x≥5orx≤− 5

− 9 <x< 3

DIRECTIONS: Write C if the inequality can be rewritten as a conjunction or D if it can be

rewritten as a disjunction. Then find the solution.

1. 
   

   |y+ 6 |> 7 2. |x+ 3 |< 4

   
   


2. 
   

   | 6 +y|≤ 2 4. | 6 y|> 12

   
   


3. 
   

   | 1 + 2 y|≥ 23 6. | 2 x− 5 |< 7

   
   


4. 
   

∣∣

∣

∣

1

4

x

∣∣

∣

∣>^12 8. |^5 −y|≤^10

CHALLENGE:Meg wrote| 2 x|<4 as a conjunction: 2x<4and2x<−4. Her

solution wasx<2andx<−2. Find her error(s).

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2011 by Judith A. Muschla, Gary Robert Muschla, and Erin Muschla. All rights reserved.