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 ³0 and
|a| = -a, if a < 0
Examples:
Solve |x| = 7
The distance between x and 0 is 7. So, x = 7 or x = -7. Solution:7 and -7
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
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.
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.
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