Answer :

GinnyAnswer
To solve the inequality [tex]\( |2x - 3| \geq 7 \)[/tex], we need to analyze the expression inside the absolute value. The absolute value inequality [tex]\( |A| \geq B \)[/tex] can be broken down into two separate inequalities: [tex]\( A \geq B \)[/tex] or [tex]\( A \leq -B \)[/tex].

Let's apply this principle to our given problem:

### Step 1: Break down the absolute value inequality

Given:
[tex]\[ |2x - 3| \geq 7 \][/tex]

This can be written as:
[tex]\[ 2x - 3 \geq 7 \quad \text{or} \quad 2x - 3 \leq -7 \][/tex]

### Step 2: Solve each inequality separately

#### Inequality 1: [tex]\( 2x - 3 \geq 7 \)[/tex]

1. Add 3 to both sides:
[tex]\[ 2x - 3 + 3 \geq 7 + 3 \][/tex]

2. Simplify:
[tex]\[ 2x \geq 10 \][/tex]

3. Divide both sides by 2:
[tex]\[ x \geq 5 \][/tex]

So one solution set is:
[tex]\[ x \geq 5 \][/tex]

#### Inequality 2: [tex]\( 2x - 3 \leq -7 \)[/tex]

1. Add 3 to both sides:
[tex]\[ 2x - 3 + 3 \leq -7 + 3 \][/tex]

2. Simplify:
[tex]\[ 2x \leq -4 \][/tex]

3. Divide both sides by 2:
[tex]\[ x \leq -2 \][/tex]

So the other solution set is:
[tex]\[ x \leq -2 \][/tex]

### Step 3: Combine the solutions

The solutions to the inequality [tex]\( |2x - 3| \geq 7 \)[/tex] are:
[tex]\[ x \geq 5 \quad \text{or} \quad x \leq -2 \][/tex]

In interval notation, the solution can be written as:
[tex]\[ (-\infty, -2] \cup [5, \infty) \][/tex]

Therefore, the solution to the inequality [tex]\( |2x - 3| \geq 7 \)[/tex] is:
[tex]\[ x \leq -2 \quad \text{or} \quad x \geq 5 \][/tex]

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