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High School Mathematics
10.3 Solve Absolute Value Equations

  • The absolute value of a number a, written as |a|, is the distance between a and 0 on a number line.
  • An absolute value equation, such as |x| = 5 is an equation that contains an absolute value expression.
  • 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.
Absolute Value
For any real number a,
|a| = a, if a 0 and
|a| = -a, if a < 0
Examples:
  1. Solve |x| = 7
    The distance between x and 0 is 7. So, x = 7 or x = -7.
    Solution:7 and -7

  2. Solve |x-4| = 9
    |x-4| = 9 -------- original equation
    Rewrite as two equations
    x-4 = 9 or x-4 = -9
    x = 13 or x = -5
    Verify:
    x = 13 x = -5
    |x-4| = 9
    |13-4| = 9
    |9| = 9
    9=9
    |x-4| = 9
    |-5-4| = 9
    |-9| = 9
    9=9
    Solution:13 and -5

  3. Solve 3|2x-7|-5 = 4
    First rewrite the equation in the form of |ax+b| = c
    3|2x-7|-5 = 4
    Adding 5 both sides
    3|2x-7|-5 +5 = 4 +5
    3|2x-7| = 9
    Divide each side by 3
    |2x-7| = 3
    Solving the absolute value equation
    |2x-7|= 3
    2x - 7 = 3 or 2x - 7 = -3
    2x = 10 or 2x = 4
    x = 5 or x = 2
    Solution: 5 and 2.

  4. Solve |3x+5|+6=-2
    |3x+5|=-8
    The absolute value of a number is never negative. So there are no solutions.
    Solution: No solution.

  5. Solve |2x+5| = 3x+4
    |2x+5| = 3x+4
    2x+5 = 3x+4 or 2x+5 = -(3x+4)
    -x = -1 or 2x+5 = -3x-4
    x = 1 or 5x = -9
    x = 1 or x = -9/5
    Check:
    |2x+5| = 3x+4 with x = 1
    |2(1)+5| = 3(1)+4
    |7| = 7
    7 = 7 True
    |2x+5| = 3x+4 with x = -9/5
    |2(-9/5)+5| = 3(-9/5)+4
    |7/5| = -7/5
    |7/5| = -7/5 Not True
    Therefore -9/5 is not a solution.
    Solution: 1

Directions: Solve the following questions. Show your work and verify the solutions. If there are not solutions write your answer as no solution. Also write at least 5 examples of your own.
Click here to view all the questions

Q 1: Solve |p - 1| + 5 = 2
Answer:

Q 2: Solve 5|6 - 5x| = 15x - 35
Answer:

Q 3: Solve |2y - 4| = -12
Answer:

Q 4: Solve |3p+7|=4
Answer:

Q 5: Solve |2x+7| = 11
Answer:

Q 6: Solve |m+3| = 7
Answer:

Q 7: Solve -3 |1 - 2/3 p| = -9
Answer:

Q 8: Solve 1/2|3x + 5| = 6x + 4
Answer:

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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