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What is the solution to the inequality [tex]|2x + 3| \ \textless \ 7[/tex]?

A. [tex]4 \ \textless \ x \ \textless \ 10[/tex]
B. [tex]-5 \ \textless \ x \ \textless \ 2[/tex]
C. [tex]x \ \textless \ 4[/tex] or [tex]x \ \textgreater \ 10[/tex]
D. [tex]x \ \textless \ -5[/tex] or [tex]x \ \textgreater \ 2[/tex]

Sagot :

GinnyAnswer
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]
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