To solve the inequality [tex]\( |2x - 3| \leq 7 \)[/tex], let's break it down into simpler parts.
The absolute value inequality [tex]\( |2x - 3| \leq 7 \)[/tex] can be rewritten as two separate inequalities:
[tex]\[ -7 \leq 2x - 3 \leq 7 \][/tex]
To solve this compound inequality, we break it into two parts and solve for [tex]\( x \)[/tex] in each part:
1. [tex]\( -7 \leq 2x - 3 \)[/tex]
2. [tex]\( 2x - 3 \leq 7 \)[/tex]
### Solving the first part: [tex]\( -7 \leq 2x - 3 \)[/tex]
Add 3 to both sides of the inequality:
[tex]\[ -7 + 3 \leq 2x \][/tex]
[tex]\[ -4 \leq 2x \][/tex]
Divide both sides by 2:
[tex]\[ -2 \leq x \][/tex]
### Solving the second part: [tex]\( 2x - 3 \leq 7 \)[/tex]
Add 3 to both sides of the inequality:
[tex]\[ 2x - 3 + 3 \leq 7 + 3 \][/tex]
[tex]\[ 2x \leq 10 \][/tex]
Divide both sides by 2:
[tex]\[ x \leq 5 \][/tex]
### Combining the results:
The solution to the inequality [tex]\( |2x - 3| \leq 7 \)[/tex] is:
[tex]\[ -2 \leq x \leq 5 \][/tex]
So, our solution set can be expressed as [tex]\( -2 \leq x \leq 5 \)[/tex].
Thus, the correct answer is:
A. [tex]\( x \geq -2 \)[/tex] and [tex]\( x \leq 5 \)[/tex]
