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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 3|2x + 6|- 9 < 15
{x: -7 < x < 1}
{x: -7 > x < 1}
{x: -7 < x > 1}
{x: -7 > x > 1 }

Q 2: |d+4| 3. Then d -1 or d -7
True
False

Q 3: If |z| 1/4. Then z _____ or z _______
-1/4, 1/4
1/4 , -1/4

Q 4: 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 5: 2 |1/4x - 5| - 4 > 3
x < 14 or x < 0
x < 6 or x > 34
x > 5 or x < 12
x > 12 or x > 38

Q 6: Solve the absolute value inequality: |x + 1| < -6
all of the others
x > 5 and x < -7
empty set
x < -7 and x > 5

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

Q 8: The inequality 1/6|2x - 1| + 2 5 the solution set is {x: x _____ or x ____}
-15/2, 9/2
-17/2, 19/2
5/3, -7/5

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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