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