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]
