Which of the following is(are) the solution(s) to [tex]|5x + 2| = 8[/tex]?

A. [tex]x = 2, -\frac{6}{5}[/tex]

B. [tex]x = \frac{6}{5}[/tex]

C. [tex]x = 2[/tex]

D. [tex]x = -2, \frac{6}{5}[/tex]



Answer :

To determine the solution(s) to the equation [tex]\( |5x + 2| = 8 \)[/tex], we need to consider the definition of the absolute value. The equation [tex]\( |A| = B \)[/tex] implies that [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Given the equation [tex]\( |5x + 2| = 8 \)[/tex], we can set up two separate equations:

1. [tex]\( 5x + 2 = 8 \)[/tex]
2. [tex]\( 5x + 2 = -8 \)[/tex]

We'll solve each of these equations separately.

Step 1: Solve [tex]\( 5x + 2 = 8 \)[/tex]

[tex]\[ 5x + 2 = 8 \][/tex]

Subtract 2 from both sides:

[tex]\[ 5x = 6 \][/tex]

Divide by 5:

[tex]\[ x = \frac{6}{5} \][/tex]

Step 2: Solve [tex]\( 5x + 2 = -8 \)[/tex]

[tex]\[ 5x + 2 = -8 \][/tex]

Subtract 2 from both sides:

[tex]\[ 5x = -10 \][/tex]

Divide by 5:

[tex]\[ x = -2 \][/tex]

Thus, the solutions to the equation [tex]\( |5x + 2| = 8 \)[/tex] are [tex]\( x = \frac{6}{5} \)[/tex] and [tex]\( x = -2 \)[/tex].

Now, let's check the given options:

- A. [tex]\( x = 2, -\frac{6}{5} \)[/tex]
- B. [tex]\( x = \frac{6}{5} \)[/tex]
- C. [tex]\( x = 2 \)[/tex]
- D. [tex]\( x = -2, \frac{6}{5} \)[/tex]

From our solution, the correct answers are [tex]\( x = \frac{6}{5} \)[/tex] and [tex]\( x = -2 \)[/tex], which matches option D.

Therefore, the correct solution is:
- D. [tex]\( x = -2, \frac{6}{5} \)[/tex]