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]
