What is the solution to [tex]\( 3|5x + 1| - 6 = 12 \)[/tex]? Choose all that apply.

A. [tex]\( x = 1 \)[/tex]

B. [tex]\( x \)[/tex] is undefined, as the equation has no solutions.

C. [tex]\( x \)[/tex] can be any real number.

D. [tex]\( x = -\frac{7}{5} \)[/tex]

E. [tex]\( x = -1 \)[/tex]

F. [tex]\( x = \frac{7}{5} \)[/tex]



Answer :

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]