Linear Inequalities Worksheets with Answer Key - Free Printable

Let's solve each of the linear inequalities by graphing and shading the correct region on the coordinate plane. I'll walk through each one step-by-step.

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🔷 General Steps for Graphing Linear Inequalities:

1. Graph the boundary line:
- If inequality is ≤ or ≥, draw a solid line.
- If it's < or >, draw a dashed line.
2. Choose a test point (usually (0,0) if not on the line).
3. Plug into the inequality to see which side to shade.
4. Shade the appropriate region.

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Let’s go through each problem:

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1. $ y < -\frac{1}{3}x + 2 $

- Boundary line: $ y = -\frac{1}{3}x + 2 $
- Slope: $ -\frac{1}{3} $, y-intercept: 2
- Draw a dashed line because it's strictly less than (<)
- Test point: (0,0)
- Plug in: $ 0 < -\frac{1}{3}(0) + 2 \Rightarrow 0 < 2 $ → True
- So shade the side containing (0,0) — below the line.

✔ Shade below the dashed line

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2. $ y \leq \frac{1}{2}x - 5 $

- Boundary line: $ y = \frac{1}{2}x - 5 $
- Slope: $ \frac{1}{2} $, y-intercept: -5
- Draw a solid line because it's ≤
- Test point: (0,0)
- $ 0 \leq \frac{1}{2}(0) - 5 \Rightarrow 0 \leq -5 $? → False
- So shade the opposite side — below the line

✔ Shade below the solid line

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3. $ y \geq \frac{1}{8}x + 2 $

- Boundary line: $ y = \frac{1}{8}x + 2 $
- Slope: $ \frac{1}{8} $, y-intercept: 2
- Draw a solid line (≥)
- Test point: (0,0)
- $ 0 \geq \frac{1}{8}(0) + 2 \Rightarrow 0 \geq 2 $? → False
- Shade the other side — above the line

✔ Shade above the solid line

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4. $ y > \frac{1}{4}x - 3 $

- Boundary line: $ y = \frac{1}{4}x - 3 $
- Slope: $ \frac{1}{4} $, y-intercept: -3
- Draw a dashed line (> )
- Test point: (0,0)
- $ 0 > \frac{1}{4}(0) - 3 \Rightarrow 0 > -3 $? → True
- So shade the side containing (0,0) — above the line

✔ Shade above the dashed line

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5. $ \frac{1}{6}x + \frac{1}{10}y \leq \frac{1}{2} $

We need to rewrite this in slope-intercept form ($ y \leq mx + b $).

#### Step 1: Eliminate fractions
Multiply both sides by the LCD of 6, 10, and 2 → LCM is 30:

$$
30 \left( \frac{1}{6}x + \frac{1}{10}y \right) \leq 30 \cdot \frac{1}{2}
$$

$$
5x + 3y \leq 15
$$

Now solve for $ y $:

$$
3y \leq -5x + 15
\Rightarrow y \leq -\frac{5}{3}x + 5
$$

- Boundary line: $ y = -\frac{5}{3}x + 5 $
- Slope: $ -\frac{5}{3} $, y-intercept: 5
- Draw solid line (≤)
- Test point: (0,0)
- $ 0 \leq -\frac{5}{3}(0) + 5 \Rightarrow 0 \leq 5 $? → True
- So shade below the line

✔ Shade below the solid line

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6. $ 5x \leq -4y - 12 $

Solve for $ y $:

$$
5x \leq -4y - 12
\Rightarrow 5x + 12 \leq -4y
\Rightarrow -4y \geq 5x + 12 \quad \text{(reverse inequality when dividing by negative)}
\Rightarrow y \leq -\frac{5}{4}x - 3
$$

Wait! Let's do that carefully:

From:
$$
5x \leq -4y - 12
\Rightarrow 5x + 12 \leq -4y
\Rightarrow -4y \geq 5x + 12 \quad \text{(reversed due to multiplying by -1)}
\Rightarrow y \leq -\frac{5}{4}x - 3
$$

Yes.

- Boundary line: $ y = -\frac{5}{4}x - 3 $
- Slope: $ -\frac{5}{4} $, y-intercept: -3
- Solid line (≤)
- Test point: (0,0)
- $ 0 \leq -\frac{5}{4}(0) - 3 \Rightarrow 0 \leq -3 $? → False
- So shade the opposite side — below the line? Wait: no!

Wait — let's check again.

We have $ y \leq -\frac{5}{4}x - 3 $

Test (0,0): Is $ 0 \leq -3 $? No → False → so (0,0) is not in solution set → shade below the line?

But wait: $ y \leq \text{negative number} $ at x=0 → y ≤ -3 → so only points with y ≤ -3 are included.

So (0,0) is above the line, and since it doesn't satisfy, we shade below the line.

✔ Shade below the solid line

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✔ Summary of All Answers:

| Problem | Inequality | Line Type | Shade |
|--------|------------|-----------|-------|
| 1 | $ y < -\frac{1}{3}x + 2 $ | Dashed | Below |
| 2 | $ y \leq \frac{1}{2}x - 5 $ | Solid | Below |
| 3 | $ y \geq \frac{1}{8}x + 2 $ | Solid | Above |
| 4 | $ y > \frac{1}{4}x - 3 $ | Dashed | Above |
| 5 | $ \frac{1}{6}x + \frac{1}{10}y \leq \frac{1}{2} $ → $ y \leq -\frac{5}{3}x + 5 $ | Solid | Below |
| 6 | $ 5x \leq -4y - 12 $ → $ y \leq -\frac{5}{4}x - 3 $ | Solid | Below |

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📌 How to Graph Each One:

For each:
1. Plot the y-intercept.
2. Use the slope to find another point.
3. Draw the line (solid or dashed).
4. Test (0,0) or pick a point not on the line.
5. Shade accordingly.

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Would you like me to generate a visual sketch or describe how to plot each line in detail?