To solve the inequality [tex]\( |3x - 3| > 12 \)[/tex], we can break it into two separate inequalities, because the absolute value of an expression [tex]\( |A| > B \)[/tex] can be interpreted as either [tex]\( A > B \)[/tex] or [tex]\( A < -B \)[/tex]. Let's proceed step-by-step:
1. Rewrite the absolute value inequality:
[tex]\[
|3x - 3| > 12
\][/tex]
This can be expressed as:
[tex]\[
3x - 3 > 12 \quad \text{or} \quad 3x - 3 < -12
\][/tex]
2. Solve the first inequality [tex]\(3x - 3 > 12\)[/tex]:
[tex]\[
3x - 3 > 12
\][/tex]
Add 3 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3x > 15
\][/tex]
Divide both sides by 3:
[tex]\[
x > 5
\][/tex]
3. Solve the second inequality [tex]\(3x - 3 < -12\)[/tex]:
[tex]\[
3x - 3 < -12
\][/tex]
Add 3 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3x < -9
\][/tex]
Divide both sides by 3:
[tex]\[
x < -3
\][/tex]
4. Combine the solutions:
The solutions to the inequality [tex]\( |3x - 3| > 12 \)[/tex] are the values of [tex]\( x \)[/tex] that satisfy either of the two inequalities derived:
[tex]\[
x > 5 \quad \text{or} \quad x < -3
\][/tex]
Therefore, the solution set for the inequality [tex]\( |3x - 3| > 12 \)[/tex] is:
[tex]\[
x \in (-\infty, -3) \cup (5, \infty)
\][/tex]
This indicates that [tex]\( x \)[/tex] must be either less than [tex]\(-3\)[/tex] or greater than [tex]\( 5 \)[/tex].
