Worksheets for inequalities - Free Printable

Problem: Solve the inequalities and plot the solution sets.

We will solve each inequality step by step and then describe how to plot the solution sets on a number line.

---

#### 1a. Solve and plot the solution set for:
\[
-10 \geq -8w - 12
\]

1. Add 12 to both sides:
\[
-10 + 12 \geq -8w - 12 + 12
\]
\[
2 \geq -8w
\]

2. Divide both sides by -8 (remember to reverse the inequality sign when dividing by a negative number):
\[
\frac{2}{-8} \leq w
\]
\[
-\frac{1}{4} \leq w
\]

3. Rewrite the inequality:
\[
w \geq -\frac{1}{4}
\]

Solution Set: \( w \geq -\frac{1}{4} \)

Plotting on a Number Line:
- Draw a closed circle at \( -\frac{1}{4} \) (since the inequality includes equality).
- Shade the region to the right of \( -\frac{1}{4} \).

---

#### 1b. Solve and plot the solution set for:
\[
\frac{r}{1} \leq -9 \cdot (-1)
\]

1. Simplify the right-hand side:
\[
\frac{r}{1} \leq 9
\]
\[
r \leq 9
\]

Solution Set: \( r \leq 9 \)

Plotting on a Number Line:
- Draw a closed circle at \( 9 \) (since the inequality includes equality).
- Shade the region to the left of \( 9 \).

---

#### 2a. Solve and plot the solution set for:
\[
-5 \geq 6n - 7
\]

1. Add 7 to both sides:
\[
-5 + 7 \geq 6n - 7 + 7
\]
\[
2 \geq 6n
\]

2. Divide both sides by 6:
\[
\frac{2}{6} \geq n
\]
\[
\frac{1}{3} \geq n
\]

3. Rewrite the inequality:
\[
n \leq \frac{1}{3}
\]

Solution Set: \( n \leq \frac{1}{3} \)

Plotting on a Number Line:
- Draw a closed circle at \( \frac{1}{3} \) (since the inequality includes equality).
- Shade the region to the left of \( \frac{1}{3} \).

---

#### 2b. Solve and plot the solution set for:
\[
-2(u - 3) \geq 10
\]

1. Distribute the -2:
\[
-2u + 6 \geq 10
\]

2. Subtract 6 from both sides:
\[
-2u + 6 - 6 \geq 10 - 6
\]
\[
-2u \geq 4
\]

3. Divide both sides by -2 (remember to reverse the inequality sign):
\[
\frac{-2u}{-2} \leq \frac{4}{-2}
\]
\[
u \leq -2
\]

Solution Set: \( u \leq -2 \)

Plotting on a Number Line:
- Draw a closed circle at \( -2 \) (since the inequality includes equality).
- Shade the region to the left of \( -2 \).

---

Final Answers:

1a. \( w \geq -\frac{1}{4} \)
1b. \( r \leq 9 \)
2a. \( n \leq \frac{1}{3} \)
2b. \( u \leq -2 \)

\[
\boxed{
\begin{aligned}
1a. & \quad w \geq -\frac{1}{4} \\
1b. & \quad r \leq 9 \\
2a. & \quad n \leq \frac{1}{3} \\
2b. & \quad u \leq -2
\end{aligned}
}
\]