To solve the inequality [tex]\( |2x - 7| + 18 \geq 30 \)[/tex], we will first isolate the absolute value expression. Here are the steps to solve the inequality:
1. Isolate the Absolute Value Expression:
[tex]\[
|2x - 7| + 18 \geq 30
\][/tex]
Subtract 18 from both sides to isolate the absolute value:
[tex]\[
|2x - 7| \geq 12
\][/tex]
2. Split the Absolute Value Inequality:
The absolute value [tex]\( |2x - 7| \geq 12 \)[/tex] can be split into two separate inequalities because [tex]\( |A| \geq B \)[/tex] implies [tex]\( A \leq -B \)[/tex] or [tex]\( A \geq B \)[/tex]:
[tex]\[
2x - 7 \leq -12 \quad \text{or} \quad 2x - 7 \geq 12
\][/tex]
3. Solve Each Inequality Separately:
- Case 1: [tex]\( 2x - 7 \leq -12 \)[/tex]
[tex]\[
2x - 7 \leq -12
\][/tex]
Add 7 to both sides:
[tex]\[
2x \leq -5
\][/tex]
Divide both sides by 2:
[tex]\[
x \leq -\frac{5}{2}
\][/tex]
- Case 2: [tex]\( 2x - 7 \geq 12 \)[/tex]
[tex]\[
2x - 7 \geq 12
\][/tex]
Add 7 to both sides:
[tex]\[
2x \geq 19
\][/tex]
Divide both sides by 2:
[tex]\[
x \geq \frac{19}{2}
\][/tex]
4. Combine the Solutions:
The solution to the inequality [tex]\( |2x - 7| + 18 \geq 30 \)[/tex] are the values of [tex]\( x \)[/tex] that satisfy either of the two inequalities derived from the absolute value:
[tex]\[
x \leq -\frac{5}{2} \quad \text{or} \quad x \geq \frac{19}{2}
\][/tex]
Thus, the final answer is:
[tex]\[
x \leq -\frac{5}{2} \text{ or } x \geq \frac{19}{2}
\][/tex]
