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High School Mathematics
10.2 Solve Absolute Value Inequalities

  • The equation |x| = 5 means that the distance between x and 0 is 5. The solutions of the equation are 5 and -5 because they are the only numbers whose distance from 0 is 5.
  • The inequality |x| < 3 means that the distance between x and 0 is less than 3, and
  • The inequality |x| > 3 means that the distance between x and 0 is greater than 3.

Solving Absolute Value Inequality:

  • The inequality |ax+b| < c where c > 0 is equivalent to the compound inequality -c < ax+b < c
  • The inequality |ax+b| > c where c > 0 is equivalent to the compound inequality ax+b < -c or ax + b > c
  • In the equalities above, < can be replaced by and > can be replaced by .
Examples:
  1. Solve |x| 8
    The solutions are x 8 and x -8

  2. Solve |x| 0.7
    |x| 0.7 -------- original equation
    Rewrite as two equations
    x 0.7 or x -0.7
    -0.7 x 0.7

  3. Solve |-4x-5|+3 < 9
    |-4x-5|+3 < 9
    subtracting 3 both sides of the inequation
    |-4x-5|+3 -3 < 9-3
    |-4x-5| < 6
    -6 < -4x-5 < 6
    adding 5 to the inequations
    -6+5 < -4x-5+5 < 6+5
    -1 < -4x < 11
    Dividing by -4 reverses the inequality sign
    0.25 > x > -2.75
    This can also be written as -2.75 < x < 0.25

  4. Solve |10 - x| > 12
    10 - x > 12 or 10 - x < -12
    • 10 -x > 12
      subtracting 10 both sides
      10 -x-10 > 12-10
      -x > 2 Multiplying both sides by (-1) changes the sign
      x < -2
    • 10 - x < -12
      subtracting 10 both sides
      10 -x-10 < -12-10
      -x < -22 Multiplying both sides by (-1) changes the sign
      x > 22
    Therefore x < -2 or x > 22

  5. |x + 2| < -1
    Since |x + 2| cannot be negative, |x + 2| cannot be less than -1. So, the solution set is the empty set.
    Solution = { }

  6. |2y -1| -4
    Since |2y - 1| is always greater than or equal to 0, the solution set is {y|y is a real number}
    The graph is the entire number line.


Directions: Solve the following questions. Also write at least 5 examples of your own. Solve them and graph the solution.
Click here for additional questions

less than or equal to greater than or equal to

Q 1: Solve the inequality |-2x + 1| > 2
{x: x < -3/2 or x > 1/2}
{x: x < -1/2 or x > 3/2}
{x: x > -1/2 or x > 3/2}
{x: x < -1 or x > 1/2}

Q 2: Solve the inequality |(x - 3)/2| + 2 < 6
{x: 5 < x < -11}
{x: -5 > x > 11}
{x: -5 < x > 11}
{x: -5 < x < 11}

Q 3: |-4x-5| + 3 < 9
-2.75 < x < 0.25
0 < x < 2
-2 < x < 5
-2.5 < x < 2.5

Q 4: |a| < 8
-8 > a < 8
-8 < a > 8
-8 > a > 8
-8 < a < 8

Q 5: Solve -3|2-5/4u| -18.
Then u ________ or u _________.
-3 1/5 , 6 2/5
2 1/2 , 5 1/3
1/2, 1/3

Q 6: The inequality |3x + 6| 12, the solution set is
{x: x____ or x__}
6 or -2
6 or 2
-6 or 2
-6 or -2

Q 7: |h| > 3.5
all of the other
-3.5 > h > 3.5
h > 3.5 or h < -3.5
h < -3.5 or h > 3.5

Q 8: Solve the absolute value inequality: |x + 4| > -3
x > -7 or x < -1
All values work
x < -1 or x > -7
All of the others

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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