Linear Inequalities Worksheets with Answer Key - Free Printable

You're asked to graph and shade each of the given linear inequalities. Here's how to do it step-by-step for each problem.

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## 📌 General Steps for Graphing a Linear Inequality:

1. Graph the boundary line:
- Treat the inequality as an equation (replace `<`, `>`, `≤`, `≥` with `=`).
- If the inequality is `≤` or `≥`, draw a solid line (includes the line).
- If it’s `<` or `>`, draw a dashed line (excludes the line).

2. Choose a test point (usually `(0,0)` if it’s not on the line) and plug into the original inequality.
- If the inequality is true → shade the side that includes the test point.
- If false → shade the opposite side.

3. Shade the correct region.

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Let’s solve each one:

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✔ 1. `y ≤ -1/3 x + 2`

- Boundary line: `y = -1/3 x + 2`
- Slope: `-1/3`, y-intercept: `(0, 2)`
- Solid line (because of `≤`)
- Test point: `(0,0)`
- Plug in: `0 ≤ -1/3(0) + 2` → `0 ≤ 2` → True
- ✔ Shade below the line (since `≤` and test point works).

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✔ 2. `y ≤ 1/2 x - 5`

- Boundary line: `y = 1/2 x - 5`
- Slope: `1/2`, y-intercept: `(0, -5)`
- Solid line (`≤`)
- Test point: `(0,0)`
- `0 ≤ 1/2(0) - 5` → `0 ≤ -5` → False
- ✔ Shade below the line (opposite of test point).

*(Note: Since the line has positive slope, “below” means the side containing points like (0,-6), not (0,0).)*

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✔ 3. `y ≥ 1/8 x + 2`

- Boundary line: `y = 1/8 x + 2`
- Slope: `1/8`, y-intercept: `(0, 2)`
- Solid line (`≥`)
- Test point: `(0,0)`
- `0 ≥ 1/8(0) + 2` → `0 ≥ 2` → False
- ✔ Shade above the line (opposite of test point).

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✔ 4. `y > 1/4 x - 3`

- Boundary line: `y = 1/4 x - 3`
- Slope: `1/4`, y-intercept: `(0, -3)`
- Dashed line (`>`)
- Test point: `(0,0)`
- `0 > 1/4(0) - 3` → `0 > -3` → True
- ✔ Shade above the line (includes test point).

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✔ 5. `1/6 x + 1/10 y ≤ 1/2`

This is in standard form. Let’s convert to slope-intercept form to graph easily.

Multiply entire inequality by 30 (LCM of 6, 10, 2) to eliminate denominators:

→ `30*(1/6 x) + 30*(1/10 y) ≤ 30*(1/2)`

→ `5x + 3y ≤ 15`

Now solve for `y`:

→ `3y ≤ -5x + 15`

→ `y ≤ (-5/3)x + 5`

✔ So now graph: `y ≤ -5/3 x + 5`

- Boundary line: `y = -5/3 x + 5`
- Slope: `-5/3`, y-intercept: `(0,5)`
- Solid line (`≤`)
- Test point: `(0,0)`
- `0 ≤ -5/3(0) + 5` → `0 ≤ 5` → True
- ✔ Shade below the line.

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✔ 6. `5x ≤ -4y - 12`

First, rearrange into slope-intercept form.

→ `5x + 4y ≤ -12`

Solve for `y`:

→ `4y ≤ -5x - 12`

→ `y ≤ (-5/4)x - 3`

✔ Graph: `y ≤ -5/4 x - 3`

- Boundary line: `y = -5/4 x - 3`
- Slope: `-5/4`, y-intercept: `(0, -3)`
- Solid line (`≤`)
- Test point: `(0,0)`
- `0 ≤ -5/4(0) - 3` → `0 ≤ -3` → False
- ✔ Shade below the line (opposite of test point).

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## 🖍️ Final Graphing Summary:

| Problem | Line Equation | Line Type | Shade Region |
|---------|---------------------|-----------|------------------|
| 1 | y = -1/3 x + 2 | Solid | Below |
| 2 | y = 1/2 x - 5 | Solid | Below |
| 3 | y = 1/8 x + 2 | Solid | Above |
| 4 | y = 1/4 x - 3 | Dashed | Above |
| 5 | y = -5/3 x + 5 | Solid | Below |
| 6 | y = -5/4 x - 3 | Solid | Below |

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## 💡 Tips for Drawing:

- Use grid lines to plot intercepts and slopes accurately.
- For fractional slopes (like 1/8, -5/3), pick x-values that make y easy to compute (e.g., for slope -5/3, go down 5, right 3 from a point).
- Always label your axes and use a ruler for straight lines.

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If you’re drawing these on paper or digitally, follow the above steps for each graph. You can also verify using online tools like Desmos.

Let me know if you’d like me to generate actual image graphs or describe specific points to plot for each line!