To solve the inequality [tex]\( |2x - 16| < 5 \)[/tex], we will follow these steps:
1. Understand the Absolute Value Inequality: The inequality [tex]\( |A| < B \)[/tex] (where [tex]\( B > 0 \)[/tex]) implies that [tex]\(-B < A < B\)[/tex]. This property will be used to remove the absolute value.
Given: [tex]\( |2x - 16| < 5 \)[/tex]
This implies:
[tex]\[ -5 < 2x - 16 < 5 \][/tex]
2. Split the Compound Inequality:
The inequality [tex]\( -5 < 2x - 16 < 5 \)[/tex] can be split into two separate inequalities:
[tex]\[ -5 < 2x - 16 \quad \text{and} \quad 2x - 16 < 5 \][/tex]
3. Solve Each Inequality Individually:
- For the first inequality:
[tex]\[ -5 < 2x - 16 \][/tex]
Add 16 to both sides:
[tex]\[ -5 + 16 < 2x \][/tex]
[tex]\[ 11 < 2x \][/tex]
Divide both sides by 2:
[tex]\[ \frac{11}{2} < x \][/tex]
[tex]\[ x > \frac{11}{2} \][/tex]
- For the second inequality:
[tex]\[ 2x - 16 < 5 \][/tex]
Add 16 to both sides:
[tex]\[ 2x < 5 + 16 \][/tex]
[tex]\[ 2x < 21 \][/tex]
Divide both sides by 2:
[tex]\[ x < \frac{21}{2} \][/tex]
[tex]\[ x < \frac{21}{2} \][/tex]
4. Combine the Results:
Combining the two parts, we get:
[tex]\[ \frac{11}{2} < x < \frac{21}{2} \][/tex]
Or equivalently:
[tex]\[ 5.5 < x < 10.5 \][/tex]
So, the solution set for the inequality [tex]\( |2x - 16| < 5 \)[/tex] is:
[tex]\[ \left( \frac{11}{2}, \frac{21}{2} \right) \][/tex]
This means [tex]\( x \)[/tex] lies in the interval [tex]\( \left( \frac{11}{2}, \frac{21}{2} \right) \)[/tex], or in decimal form, [tex]\( (5.5, 10.5) \)[/tex].
