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### High School Mathematics10.3 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 ³0 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
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.

 Q 1: Solve |m+3| = 7Answer: Q 2: Solve |3p+7|=4Answer: Q 3: Solve |2x+7| = 11Answer: Q 4: Solve 3|13 - 2y| = 15Answer: Q 5: Solve |p - 1| + 5 = 2Answer: Q 6: Solve -3 |1 - 2/3 p| = -9Answer: Q 7: Solve -9 |4p+2| - 8 = -35Answer: Q 8: Solve -10 |14 - r| - 2 = -7Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!