To solve the equation [tex]\( |6x + 3| = 21 \)[/tex], we need to consider the definition of absolute value. The absolute value of an expression can be described as follows:
[tex]\[ |A| = B \Rightarrow A = B \text{ or } A = -B \][/tex]
Given the equation [tex]\( |6x + 3| = 21 \)[/tex], we can set up two separate equations based on the definition of absolute value:
1. [tex]\( 6x + 3 = 21 \)[/tex]
2. [tex]\( 6x + 3 = -21 \)[/tex]
We will solve each of these equations separately.
### Solving [tex]\( 6x + 3 = 21 \)[/tex]:
1. Subtract 3 from both sides:
[tex]\[ 6x = 21 - 3 \][/tex]
[tex]\[ 6x = 18 \][/tex]
2. Divide both sides by 6:
[tex]\[ x = \frac{18}{6} \][/tex]
[tex]\[ x = 3 \][/tex]
### Solving [tex]\( 6x + 3 = -21 \)[/tex]:
1. Subtract 3 from both sides:
[tex]\[ 6x = -21 - 3 \][/tex]
[tex]\[ 6x = -24 \][/tex]
2. Divide both sides by 6:
[tex]\[ x = \frac{-24}{6} \][/tex]
[tex]\[ x = -4 \][/tex]
Thus, the solutions to the equation [tex]\( |6x + 3| = 21 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -4 \)[/tex].
So, the correct answer is:
C. [tex]\( x = 3 \)[/tex] and [tex]\( x = -4 \)[/tex]
