To solve the absolute value equation [tex]\(3|x - 4| = 33\)[/tex], follow these steps:
1. Isolate the absolute value expression:
Divide both sides of the equation by 3:
[tex]\[
|x - 4| = \frac{33}{3}
\][/tex]
Simplifying the right side, we get:
[tex]\[
|x - 4| = 11
\][/tex]
2. Set up two separate equations:
Since the absolute value expression [tex]\( |x - 4| \)[/tex] equals 11, we need to consider both the positive and negative cases for the expression inside the absolute value. This gives us two equations to solve:
[tex]\[
x - 4 = 11
\][/tex]
[tex]\[
x - 4 = -11
\][/tex]
3. Solve each equation separately:
- For the first equation [tex]\( x - 4 = 11 \)[/tex]:
Add 4 to both sides:
[tex]\[
x = 11 + 4
\][/tex]
Simplifying, we find:
[tex]\[
x = 15
\][/tex]
- For the second equation [tex]\( x - 4 = -11 \)[/tex]:
Add 4 to both sides:
[tex]\[
x = -11 + 4
\][/tex]
Simplifying, we find:
[tex]\[
x = -7
\][/tex]
4. Combine the solutions:
The values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 3|x - 4| = 33 \)[/tex] are:
[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]
Therefore, the solutions to the equation are:
[tex]\[
x = 15 \quad \text{and} \quad x = -7
\][/tex]
