To solve the inequality \(|2x - 3| \leq 11\), we need to break it down into two separate inequalities based on the definition of absolute value.
The inequality \(|A| \leq B\) implies that \(-B \leq A \leq B\). Therefore, we can rewrite our inequality as:
\[ -11 \leq 2x - 3 \leq 11 \]
We will solve each side of this compound inequality separately.
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Solve the left side: \[ -11 \leq 2x - 3 \]
Add 3 to both sides: \[ -11 + 3 \leq 2x \] \[ -8 \leq 2x \]
Divide by 2: \[ -4 \leq x \quad \text{or} \quad x \geq -4 \]
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Solve the right side: \[ 2x - 3 \leq 11 \]
Add 3 to both sides: \[ 2x \leq 11 + 3 \] \[ 2x \leq 14 \]
Divide by 2: \[ x \leq 7 \]
Now, we combine the two parts of the solution:
\[ -4 \leq x \leq 7 \]
This can be written in interval notation as:
\[ [-4, 7] \]
So the solution set of the inequality \(|2x - 3| \leq 11\) is:
Answer: \([-4, 7]\) (set notation).
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