Math worksheet for solving inequalities with equations and variables.

Solving Inequalities Worksheet

Here is the step-by-step solution for each inequality problem from the worksheet. I will solve them one by one and explain the process.

---

#### 1. \( 6(n + 10) \geq 31 \)

1. Distribute the 6:
\[
6n + 60 \geq 31
\]

2. Subtract 60 from both sides:
\[
6n \geq 31 - 60
\]
\[
6n \geq -29
\]

3. Divide both sides by 6:
\[
n \geq \frac{-29}{6}
\]

Solution:
\[
n \geq -\frac{29}{6}
\]

---

#### 2. \( 6(y + 6) \geq 35 \)

1. Distribute the 6:
\[
6y + 36 \geq 35
\]

2. Subtract 36 from both sides:
\[
6y \geq 35 - 36
\]
\[
6y \geq -1
\]

3. Divide both sides by 6:
\[
y \geq \frac{-1}{6}
\]

Solution:
\[
y \geq -\frac{1}{6}
\]

---

#### 3. \( -8 \leq \frac{m}{10} - 10 \)

1. Add 10 to both sides:
\[
-8 + 10 \leq \frac{m}{10}
\]
\[
2 \leq \frac{m}{10}
\]

2. Multiply both sides by 10:
\[
2 \times 10 \leq m
\]
\[
20 \leq m
\]

Solution:
\[
m \geq 20
\]

---

#### 4. \( 5 \leq \frac{u}{-3} + 19 \)

1. Subtract 19 from both sides:
\[
5 - 19 \leq \frac{u}{-3}
\]
\[
-14 \leq \frac{u}{-3}
\]

2. Multiply both sides by -3 (and reverse the inequality sign):
\[
-14 \times (-3) \geq u
\]
\[
42 \geq u
\]

Solution:
\[
u \leq 42
\]

---

#### 5. \( 6r + 10 \geq 19 \)

1. Subtract 10 from both sides:
\[
6r \geq 19 - 10
\]
\[
6r \geq 9
\]

2. Divide both sides by 6:
\[
r \geq \frac{9}{6}
\]
\[
r \geq \frac{3}{2}
\]

Solution:
\[
r \geq \frac{3}{2}
\]

---

#### 6. \( 7g + 18 < 4g + 10 \)

1. Subtract \( 4g \) from both sides:
\[
7g - 4g + 18 < 10
\]
\[
3g + 18 < 10
\]

2. Subtract 18 from both sides:
\[
3g < 10 - 18
\]
\[
3g < -8
\]

3. Divide both sides by 3:
\[
g < \frac{-8}{3}
\]

Solution:
\[
g < -\frac{8}{3}
\]

---

#### 7. \( 5m + 2 < 9m + 1 \)

1. Subtract \( 5m \) from both sides:
\[
2 < 9m - 5m + 1
\]
\[
2 < 4m + 1
\]

2. Subtract 1 from both sides:
\[
2 - 1 < 4m
\]
\[
1 < 4m
\]

3. Divide both sides by 4:
\[
\frac{1}{4} < m
\]

Solution:
\[
m > \frac{1}{4}
\]

---

#### 8. \( -21 > -2m + 7 \)

1. Subtract 7 from both sides:
\[
-21 - 7 > -2m
\]
\[
-28 > -2m
\]

2. Divide both sides by -2 (and reverse the inequality sign):
\[
\frac{-28}{-2} < m
\]
\[
14 < m
\]

Solution:
\[
m > 14
\]

---

#### 9. \( 58 > 14 + 11w \)

1. Subtract 14 from both sides:
\[
58 - 14 > 11w
\]
\[
44 > 11w
\]

2. Divide both sides by 11:
\[
\frac{44}{11} > w
\]
\[
4 > w
\]

Solution:
\[
w < 4
\]

---

#### 10. \( -5(u - 19) \leq -6 + 2u \)

1. Distribute the -5:
\[
-5u + 95 \leq -6 + 2u
\]

2. Add \( 5u \) to both sides:
\[
95 \leq -6 + 2u + 5u
\]
\[
95 \leq -6 + 7u
\]

3. Add 6 to both sides:
\[
95 + 6 \leq 7u
\]
\[
101 \leq 7u
\]

4. Divide both sides by 7:
\[
\frac{101}{7} \leq u
\]

Solution:
\[
u \geq \frac{101}{7}
\]

---

#### 11. \( 4h - 6 \geq 140 \)

1. Add 6 to both sides:
\[
4h \geq 140 + 6
\]
\[
4h \geq 146
\]

2. Divide both sides by 4:
\[
h \geq \frac{146}{4}
\]
\[
h \geq 36.5
\]

Solution:
\[
h \geq 36.5
\]

---

#### 12. \( 144 \leq 12a - 4 \)

1. Add 4 to both sides:
\[
144 + 4 \leq 12a
\]
\[
148 \leq 12a
\]

2. Divide both sides by 12:
\[
\frac{148}{12} \leq a
\]
\[
\frac{37}{3} \leq a
\]

Solution:
\[
a \geq \frac{37}{3}
\]

---

#### 13. \( 22 \leq -7(b - 19) \)

1. Distribute the -7:
\[
22 \leq -7b + 133
\]

2. Subtract 133 from both sides:
\[
22 - 133 \leq -7b
\]
\[
-111 \leq -7b
\]

3. Divide both sides by -7 (and reverse the inequality sign):
\[
\frac{-111}{-7} \geq b
\]
\[
\frac{111}{7} \geq b
\]

Solution:
\[
b \leq \frac{111}{7}
\]

---

#### 14. \( 6u + 6 < u + 29 \)

1. Subtract \( u \) from both sides:
\[
6u - u + 6 < 29
\]
\[
5u + 6 < 29
\]

2. Subtract 6 from both sides:
\[
5u < 29 - 6
\]
\[
5u < 23
\]

3. Divide both sides by 5:
\[
u < \frac{23}{5}
\]

Solution:
\[
u < \frac{23}{5}
\]

---

Final Answer Key

\[
\boxed{
\begin{aligned}
1. & \quad n \geq -\frac{29}{6} \\
2. & \quad y \geq -\frac{1}{6} \\
3. & \quad m \geq 20 \\
4. & \quad u \leq 42 \\
5. & \quad r \geq \frac{3}{2} \\
6. & \quad g < -\frac{8}{3} \\
7. & \quad m > \frac{1}{4} \\
8. & \quad m > 14 \\
9. & \quad w < 4 \\
10. & \quad u \geq \frac{101}{7} \\
11. & \quad h \geq 36.5 \\
12. & \quad a \geq \frac{37}{3} \\
13. & \quad b \leq \frac{111}{7} \\
14. & \quad u < \frac{23}{5}
\end{aligned}
}
\]