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

1. |2x - 3| /leq 11
2. 50 > 7n + 8 > 22
3. |2w - 5| /geq 0
4. 12 > |4d + 16|
5. -5n < -n + 3 < 10
6. |1.5t - 0.75| < 4

1 answer

Let's solve each inequality step by step.

1. \(|2x - 3| \leq 11\)

This absolute value inequality can be split into two separate inequalities:

\[ -11 \leq 2x - 3 \leq 11 \]

First part: \[ 2x - 3 \geq -11 \] \[ 2x \geq -8 \] \[ x \geq -4 \]

Second part: \[ 2x - 3 \leq 11 \] \[ 2x \leq 14 \] \[ x \leq 7 \]

Final solution: \[ x \in [-4, 7] \]

2. \(50 > 7n + 8 > 22\)

This compound inequality can be solved in two parts.

Left part: \[ 50 > 7n + 8 \] \[ 42 > 7n \] \[ 6 > n \quad \text{or} \quad n < 6 \]

Right part: \[ 7n + 8 > 22 \] \[ 7n > 14 \] \[ n > 2 \]

Final solution: \[ n \in (2, 6) \]

3. \(|2w - 5| \geq 0\)

The absolute value is always greater than or equal to zero. Thus:

Final solution: \[ w \in \mathbb{R} \quad \text{(all real numbers)} \]

4. \(12 > |4d + 16|\)

This absolute value inequality can be split into two inequalities:

\[ -12 < 4d + 16 < 12 \]

Left part: \[ 4d + 16 > -12 \] \[ 4d > -28 \] \[ d > -7 \]

Right part: \[ 4d + 16 < 12 \] \[ 4d < -4 \] \[ d < -1 \]

Final solution: \[ d \in (-7, -1) \]

5. \(-5n < -n + 3 < 10\)

This compound inequality can be solved in two parts.

Left part: \[ -5n < -n + 3 \] \[ -4n < 3 \] \[ n > -\frac{3}{4} \quad \text{(reversing the inequality)} \]

Right part: \[ -n + 3 < 10 \] \[ -n < 7 \] \[ n > -7 \]

Thus the intersection of the two results:

Final solution: \[ n \in (-\frac{3}{4}, 7) \]

6. \(|1.5t - 0.75| < 4\)

This absolute value inequality can be split into two inequalities:

\[ -4 < 1.5t - 0.75 < 4 \]

Left part: \[ 1.5t - 0.75 > -4 \] \[ 1.5t > -3.25 \] \[ t > -\frac{13}{6} \]

Right part: \[ 1.5t - 0.75 < 4 \] \[ 1.5t < 4.75 \] \[ t < \frac{19}{6} \]

Final solution: \[ t \in \left(-\frac{13}{6}, \frac{19}{6}\right) \]


Summary of Solutions:

  1. \(x \in [-4, 7]\)
  2. \(n \in (2, 6)\)
  3. \(w \in \mathbb{R}\)
  4. \(d \in (-7, -1)\)
  5. \(n \in (-\frac{3}{4}, 7)\)
  6. \(t \in \left(-\frac{13}{6}, \frac{19}{6}\right)\)