Certainly! Let's solve the given inequality [tex]\( |2x + 3| < 7 \)[/tex] step-by-step.
### Step 1: Understand the Inequality
The absolute value inequality [tex]\( |2x + 3| < 7 \)[/tex] can be broken down into two separate linear inequalities:
[tex]\[ -7 < 2x + 3 < 7 \][/tex]
### Step 2: Break it Down
We can express [tex]\( |2x + 3| < 7 \)[/tex] as a compound inequality:
[tex]\[ -7 < 2x + 3 \quad \text{and} \quad 2x + 3 < 7 \][/tex]
### Step 3: Solve the Compound Inequality
#### Part 1: Solve [tex]\( -7 < 2x + 3 \)[/tex]
First, isolate [tex]\( x \)[/tex]:
1. Subtract 3 from both sides:
[tex]\[ -7 - 3 < 2x \][/tex]
[tex]\[ -10 < 2x \][/tex]
2. Divide both sides by 2:
[tex]\[ -5 < x \][/tex]
#### Part 2: Solve [tex]\( 2x + 3 < 7 \)[/tex]
Next, isolate [tex]\( x \)[/tex]:
1. Subtract 3 from both sides:
[tex]\[ 2x < 7 - 3 \][/tex]
[tex]\[ 2x < 4 \][/tex]
2. Divide both sides by 2:
[tex]\[ x < 2 \][/tex]
### Step 4: Combine the Solutions
From both parts, we obtained:
[tex]\[ -5 < x \][/tex]
[tex]\[ x < 2 \][/tex]
Combining these results, we get:
[tex]\[ -5 < x < 2 \][/tex]
### Step 5: Write the Final Solution
Therefore, the solution to the inequality [tex]\( |2x + 3| < 7 \)[/tex] is:
[tex]\[ -5 < x < 2 \][/tex]
### Step 6: Find the Correct Choice
Looking at the given options, the correct choice is:
[tex]\[ \boxed{-5 < x < 2} \][/tex]
