To solve the equation [tex]\(3|5x + 1| - 6 = 12\)[/tex], we'll follow these steps:
1. Isolate the absolute value term:
Start by adding 6 to both sides of the equation:
[tex]\[
3|5x + 1| - 6 + 6 = 12 + 6
\][/tex]
This simplifies to:
[tex]\[
3|5x + 1| = 18
\][/tex]
2. Divide by 3 to further isolate the absolute value:
[tex]\[
\frac{3|5x + 1|}{3} = \frac{18}{3}
\][/tex]
This simplifies to:
[tex]\[
|5x + 1| = 6
\][/tex]
3. Consider the definition of the absolute value:
The equation [tex]\(|5x + 1| = 6\)[/tex] means [tex]\(5x + 1\)[/tex] could be either 6 or -6. Thus, we have two cases to consider.
Case 1:
[tex]\[
5x + 1 = 6
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
5x = 6 - 1
\][/tex]
[tex]\[
5x = 5
\][/tex]
[tex]\[
x = 1
\][/tex]
Case 2:
[tex]\[
5x + 1 = -6
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
5x = -6 - 1
\][/tex]
[tex]\[
5x = -7
\][/tex]
[tex]\[
x = -\frac{7}{5}
\][/tex]
4. Check the solutions:
Both [tex]\(x = 1\)[/tex] and [tex]\(x = -\frac{7}{5}\)[/tex] should be checked to confirm they satisfy the original equation:
For [tex]\(x = 1\)[/tex]:
[tex]\[
3|5(1) + 1| - 6 = 3|5 + 1| - 6 = 3|6| - 6 = 3 \times 6 - 6 = 18 - 6 = 12 \quad \text{(correct!)}
\][/tex]
For [tex]\(x = -\frac{7}{5}\)[/tex]:
[tex]\[
3|5\left(-\frac{7}{5}\right) + 1| - 6 = 3| -7 + 1| - 6 = 3| -6| - 6 = 3 \times 6 - 6 = 18 - 6 = 12 \quad \text{(correct!)}
\][/tex]
The solutions are [tex]\(x = 1\)[/tex] and [tex]\(x = -\frac{7}{5}\)[/tex].
Therefore, the correct options are:
- A. [tex]\(x = 1\)[/tex]
- D. [tex]\(x=-\frac{7}{5}\)[/tex]
