To find the solution set of the compound inequality [tex]\(3x + 2 \leq 5x + 6 \leq 6x + 7\)[/tex], we need to break it down into two individual inequalities and solve each one separately. Let's break it into two separate inequalities:
1. [tex]\(3x + 2 \leq 5x + 6\)[/tex]
2. [tex]\(5x + 6 \leq 6x + 7\)[/tex]
### Solve the first inequality [tex]\(3x + 2 \leq 5x + 6\)[/tex]:
1. Start by subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
2 \leq 2x + 6
\][/tex]
2. Next, subtract 6 from both sides:
[tex]\[
2 - 6 \leq 2x
\][/tex]
[tex]\[
-4 \leq 2x
\][/tex]
3. Finally, divide both sides by 2:
[tex]\[
-2 \leq x
\][/tex]
This simplifies to:
[tex]\[
x \geq -2
\][/tex]
### Solve the second inequality [tex]\(5x + 6 \leq 6x + 7\)[/tex]:
1. Start by subtracting [tex]\(5x\)[/tex] from both sides:
[tex]\[
6 \leq x + 7
\][/tex]
2. Next, subtract 7 from both sides:
[tex]\[
6 - 7 \leq x
\][/tex]
[tex]\[
-1 \leq x
\][/tex]
This simplifies to:
[tex]\[
x \geq -1
\][/tex]
### Combine the solutions of the two inequalities:
- From the first inequality, we have [tex]\(x \geq -2\)[/tex].
- From the second inequality, we have [tex]\(x \geq -1\)[/tex].
Since [tex]\(x \geq -1\)[/tex] is a stricter constraint than [tex]\(x \geq -2\)[/tex], the solution to the compound inequality is [tex]\(x \geq -1\)[/tex].
Therefore, the solution set of [tex]\(3x + 2 \leq 5x + 6 \leq 6x + 7\)[/tex] is [tex]\(x \geq -1\)[/tex].
