To solve the equation [tex]\(\frac{1}{3}|3x + 9| - 5 = 4\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here is the step-by-step approach:
1. Start with the given equation:
[tex]\[
\frac{1}{3}|3x + 9| - 5 = 4
\][/tex]
2. Isolate the absolute value term:
To do that, first add 5 to both sides:
[tex]\[
\frac{1}{3}|3x + 9| = 9
\][/tex]
3. Eliminate the fraction:
Multiply both sides by 3 to get rid of the fraction:
[tex]\[
|3x + 9| = 27
\][/tex]
4. Solve the absolute value equation:
The equation [tex]\(|3x + 9| = 27\)[/tex] can be split into two separate equations:
[tex]\[
3x + 9 = 27 \quad \text{and} \quad 3x + 9 = -27
\][/tex]
5. Solve each equation separately:
For [tex]\(3x + 9 = 27\)[/tex]:
[tex]\[
3x + 9 = 27
\][/tex]
Subtract 9 from both sides:
[tex]\[
3x = 18
\][/tex]
Divide both sides by 3:
[tex]\[
x = 6
\][/tex]
For [tex]\(3x + 9 = -27\)[/tex]:
[tex]\[
3x + 9 = -27
\][/tex]
Subtract 9 from both sides:
[tex]\[
3x = -36
\][/tex]
Divide both sides by 3:
[tex]\[
x = -12
\][/tex]
6. Verify the solutions:
We have found [tex]\(x = 6\)[/tex] and [tex]\(x = -12\)[/tex]. To verify, substitute these values back into the original equation:
For [tex]\(x = 6\)[/tex]:
[tex]\[
\frac{1}{3}|3(6) + 9| - 5 = \frac{1}{3}|18 + 9| - 5 = \frac{1}{3}|27| - 5 = 9 - 5 = 4
\][/tex]
This checks out.
For [tex]\(x = -12\)[/tex]:
[tex]\[
\frac{1}{3}|3(-12) + 9| - 5 = \frac{1}{3}|-36 + 9| - 5 = \frac{1}{3}|-27| - 5 = 9 - 5 = 4
\][/tex]
This also checks out.
Therefore, the solution to the equation [tex]\(\frac{1}{3}|3x + 9| - 5 = 4\)[/tex] is:
[tex]\[
x = 6 \quad \text{or} \quad x = -12
\][/tex]
The correct answer is:
[tex]\[
x = 6 \text{ or } x = -12
\][/tex]
