To solve the compound inequality [tex]\( -7 \leq 6x + 5 \leq 23 \)[/tex] for [tex]\( x \)[/tex], we need to break it down into two separate inequalities and solve each one individually.
1. Solving the lower bound inequality:
[tex]\[
-7 \leq 6x + 5
\][/tex]
- Subtract 5 from both sides:
[tex]\[
-7 - 5 \leq 6x
\][/tex]
- Simplify the left-hand side:
[tex]\[
-12 \leq 6x
\][/tex]
- Divide by 6 to isolate [tex]\( x \)[/tex]:
[tex]\[
-12 / 6 \leq x
\][/tex]
- Simplify the division:
[tex]\[
-2 \leq x
\][/tex]
2. Solving the upper bound inequality:
[tex]\[
6x + 5 \leq 23
\][/tex]
- Subtract 5 from both sides:
[tex]\[
6x \leq 23 - 5
\][/tex]
- Simplify the right-hand side:
[tex]\[
6x \leq 18
\][/tex]
- Divide by 6 to isolate [tex]\( x \)[/tex]:
[tex]\[
x \leq 18 / 6
\][/tex]
- Simplify the division:
[tex]\[
x \leq 3
\][/tex]
3. Combining the results:
From the lower bound, we have [tex]\( -2 \leq x \)[/tex]. From the upper bound, we have [tex]\( x \leq 3 \)[/tex]. Combining these two results gives us the solution:
[tex]\[
-2 \leq x \leq 3
\][/tex]
Therefore, the correct answer is:
d. [tex]\( -2 \leq x \leq 3 \)[/tex]
