Question

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

Answer 1

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. \(t \in \left(-\frac{13}{6}, \frac{19}{6}\right)\)