Question

solving absolute value equations
steps:
1 isolate the absolute value.
2 create two equations by setting the “insides” of the absolute value equal to the positive and the negative version of the number on the other side of the equation.
3 solve both equations.
4 write your answer in a solution set. x = { , }
examples:

1. |x| = 3
2. |p| = 12

p = - 12 p = 12
p = { - 12, 12 }

1. |8y| = 40
2. | - 3z| = 27

| - 3z| = 27

1. |x - 2| = 4
2. |q + 9| = 11

Explanation:

Step1: Solve |x| = 3

Set $x = 3$ and $x=-3$, so the solution set is $x=\{-3,3\}$.

Step2: Solve |p| = 12

Set $p = 12$ and $p = - 12$, so the solution set is $p=\{-12,12\}$.

Step3: Solve |8y| = 40

First, isolate the variable: $|y|=\frac{40}{8}=5$. Then set $y = 5$ and $y=-5$, so the solution set is $y=\{-5,5\}$.

Step4: Solve |-3z| = 27

Since $|-3z| = 3|z|$, we have $|z|=\frac{27}{3}=9$. Set $z = 9$ and $z=-9$, so the solution set is $z=\{-9,9\}$.

Step5: Solve |x - 2| = 4

Set $x-2 = 4$ and $x - 2=-4$. For $x-2 = 4$, $x=6$; for $x - 2=-4$, $x=-2$. So the solution set is $x=\{-2,6\}$.

Step6: Solve |q + 9| = 11

Set $q + 9=11$ and $q + 9=-11$. For $q + 9=11$, $q=2$; for $q + 9=-11$, $q=-20$. So the solution set is $q=\{-20,2\}$.

Answer:

1. $x=\{-3,3\}$
2. $p=\{-12,12\}$
3. $y=\{-5,5\}$
4. $z=\{-9,9\}$
5. $x=\{-2,6\}$
6. $q=\{-20,2\}$