Solve each inequality. Write the solutions as either the union or intersection of two sets.

1. |2x - 3| /leq 11

1 answer

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.

  1. Solve the left side: \[ -11 \leq 2x - 3 \]

    Add 3 to both sides: \[ -11 + 3 \leq 2x \] \[ -8 \leq 2x \]

    Divide by 2: \[ -4 \leq x \quad \text{or} \quad x \geq -4 \]

  2. 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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