Let's solve the compound inequality [tex]\(1 \leq 2x + 6 < 15\)[/tex]:
### Step 1: Break down the compound inequality into two separate inequalities.
We have:
1. [tex]\(1 \leq 2x + 6\)[/tex]
2. [tex]\(2x + 6 < 15\)[/tex]
### Step 2: Solve each inequality separately.
#### Solve [tex]\(1 \leq 2x + 6\)[/tex]:
1. Subtract 6 from both sides:
[tex]\[ 1 - 6 \leq 2x + 6 - 6 \][/tex]
[tex]\[ -5 \leq 2x \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{-5}{2} \leq x \][/tex]
[tex]\[ -2.5 \leq x \][/tex]
This simplifies to:
[tex]\[ x \geq -2.5 \][/tex]
#### Solve [tex]\(2x + 6 < 15\)[/tex]:
1. Subtract 6 from both sides:
[tex]\[ 2x + 6 - 6 < 15 - 6 \][/tex]
[tex]\[ 2x < 9 \][/tex]
2. Divide both sides by 2:
[tex]\[ \frac{2x}{2} < \frac{9}{2} \][/tex]
[tex]\[ x < 4.5 \][/tex]
### Step 3: Combine the solutions.
From the two inequalities [tex]\(x \geq -2.5\)[/tex] and [tex]\(x < 4.5\)[/tex], we can combine them to form the compound inequality:
[tex]\[ -2.5 \leq x < 4.5 \][/tex]
### Final solution:
The solution set for the inequality [tex]\(1 \leq 2x + 6 < 15\)[/tex] is:
[tex]\[ \boxed{-2.5 \leq x < 4.5} \][/tex]
