To solve the equation [tex]\( |3x - 8| = 22 \)[/tex], we need to consider the definition of absolute value, which requires us to examine two possible cases:
1. [tex]\( 3x - 8 = 22 \)[/tex]
2. [tex]\( 3x - 8 = -22 \)[/tex]
Let's solve each case separately:
Case 1: [tex]\( 3x - 8 = 22 \)[/tex]
1. Add 8 to both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3x - 8 + 8 = 22 + 8
\][/tex]
[tex]\[
3x = 30
\][/tex]
2. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{30}{3}
\][/tex]
[tex]\[
x = 10
\][/tex]
Case 2: [tex]\( 3x - 8 = -22 \)[/tex]
1. Add 8 to both sides of the equation:
[tex]\[
3x - 8 + 8 = -22 + 8
\][/tex]
[tex]\[
3x = -14
\][/tex]
2. Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-14}{3}
\][/tex]
[tex]\[
x = -\frac{14}{3}
\][/tex]
Now, listing both solutions, we get:
[tex]\[
x = 10 \quad \text{or} \quad x = -\frac{14}{3}
\][/tex]
Looking at the given multiple-choice options:
- [tex]\( x = -\frac{14}{3} \)[/tex] or [tex]\( x = 10 \)[/tex]
- [tex]\( x = \frac{14}{3} \)[/tex] or [tex]\( x = 10 \)[/tex]
- [tex]\( x = -10 \)[/tex] or [tex]\( x = 10 \)[/tex]
- [tex]\( x = 10 \)[/tex]
The correct answer is the first option:
[tex]\[
x = -\frac{14}{3} \quad \text{or} \quad x = 10
\][/tex]
So the solution is:
[tex]\[ x = -\frac{14}{3} \text{ or } x = 10 \][/tex]
