To solve the compound inequality [tex]\(1 < 2x + 1 < 9\)[/tex], we need to break it down into two separate inequalities and solve each one step by step.
### Step 1: Solve the Left Inequality
First, consider the inequality:
[tex]\[ 1 < 2x + 1 \][/tex]
1. Subtract 1 from both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[ 1 - 1 < 2x + 1 - 1 \][/tex]
[tex]\[ 0 < 2x \][/tex]
2. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{0}{2} < \frac{2x}{2} \][/tex]
[tex]\[ 0 < x \][/tex]
### Step 2: Solve the Right Inequality
Next, consider the inequality:
[tex]\[ 2x + 1 < 9 \][/tex]
1. Subtract 1 from both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[ 2x + 1 - 1 < 9 - 1 \][/tex]
[tex]\[ 2x < 8 \][/tex]
2. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{2x}{2} < \frac{8}{2} \][/tex]
[tex]\[ x < 4 \][/tex]
### Step 3: Combine the Results
Combining the solutions of the two parts, we get:
[tex]\[ 0 < x < 4 \][/tex]
Therefore, the solution to the compound inequality [tex]\(1 < 2x + 1 < 9\)[/tex] is:
[tex]\[ 0 < x < 4 \][/tex]
