Graphing Systems of Linear Inequalities worksheet - Free Printable

Let's go through each of the four systems of inequalities and analyze the graphs provided to verify whether the shaded regions correctly represent the solution sets. Then, I’ll explain how to solve each system step by step.

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🔷 Problem 1:

Inequalities:
- $ y \leq -x - 2 $
- $ y \geq -5x + 2 $

#### ✔ Step-by-step Solution:
1. Graph the boundary lines:
- $ y = -x - 2 $: Solid line (because of ≤)
- $ y = -5x + 2 $: Solid line (because of ≥)

2. Determine shading:
- For $ y \leq -x - 2 $: Shade below the line.
- For $ y \geq -5x + 2 $: Shade above the line.

3. Find the overlapping region: The solution is where both shaded areas overlap.

4. Check the graph:
- The two lines intersect somewhere in the second quadrant.
- The shaded region is below the first line and above the second line.
- The red shaded area appears correct — it lies between the two lines and satisfies both inequalities.

✔ Conclusion: Graph 1 is correct.

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🔷 Problem 2:

Inequalities:
- $ y > -x - 2 $
- $ y < -5x + 2 $

#### ✔ Step-by-step Solution:
1. Boundary lines:
- $ y = -x - 2 $: Dashed line (because of >)
- $ y = -5x + 2 $: Dashed line (because of <)

2. Shading:
- $ y > -x - 2 $: Shade above the dashed line.
- $ y < -5x + 2 $: Shade below the dashed line.

3. Overlap: The solution is the region between the two lines, not including the lines themselves.

4. Check the graph:
- Both lines are dashed (correct).
- Shading is between them — this matches.
- The region is shaded in red and bounded by the two lines.

✔ Conclusion: Graph 2 is correct.

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🔷 Problem 3:

Inequalities:
- $ y \leq \frac{1}{2}x + 2 $
- $ y < -2x - 3 $

#### ✔ Step-by-step Solution:
1. Boundary lines:
- $ y = \frac{1}{2}x + 2 $: Solid line (≤)
- $ y = -2x - 3 $: Dashed line (<)

2. Shading:
- $ y \leq \frac{1}{2}x + 2 $: Shade below or on the solid line.
- $ y < -2x - 3 $: Shade below the dashed line.

3. Overlap: Find where both conditions hold.

4. Check the graph:
- One line has a positive slope ($ \frac{1}{2} $), the other negative (-2).
- The shaded region is below both lines.
- The intersection point appears around $ x = -2 $, $ y = -1 $, which can be checked:
- Plug $ x = -2 $ into both:
- $ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 $
- $ y = -2(-2) - 3 = 4 - 3 = 1 $
- So they intersect at $ (-2, 1) $
- Below both lines → shaded region should be below both, forming a wedge pointing downward.
- The graph shows a shaded region in the lower-left corner, bounded by both lines.

✔ Conclusion: Graph 3 is correct.

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🔷 Problem 4:

Inequalities:
- $ x \leq -3 $
- $ y < \frac{5}{3}x + 2 $

#### ✔ Step-by-step Solution:
1. Boundary lines:
- $ x = -3 $: Vertical line, solid (because of ≤)
- $ y = \frac{5}{3}x + 2 $: Dashed line (because of <)

2. Shading:
- $ x \leq -3 $: Shade to the left of the vertical line.
- $ y < \frac{5}{3}x + 2 $: Shade below the line.

3. Overlap: The solution is the region that is:
- To the left of $ x = -3 $
- And below the line $ y = \frac{5}{3}x + 2 $

4. Check the graph:
- Vertical line at $ x = -3 $: solid → correct.
- Line $ y = \frac{5}{3}x + 2 $: dashed → correct.
- Shaded region is to the left of $ x = -3 $ and below the slanted line.
- The shaded area is in the bottom-left quadrant, extending down and left.

✔ Conclusion: Graph 4 is correct.

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✔ Final Summary:

All four graphs are correctly sketched and represent the solution sets for their respective systems of inequalities.

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📌 Key Concepts Recap:

| Inequality | Line Type | Shading |
|-----------|----------|--------|
| $ y \leq mx + b $ | Solid | Below or on |
| $ y \geq mx + b $ | Solid | Above or on |
| $ y < mx + b $ | Dashed | Below |
| $ y > mx + b $ | Dashed | Above |
| $ x \leq c $ | Solid vertical line | Left of line |
| $ x > c $ | Dashed vertical line | Right of line |

> The solution to a system is the intersection of all shaded regions.

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✔ Answer: All four graphs are correct representations of their respective systems of inequalities.