Systems of Inequalities Worksheets with Answer Key - Free Printable

Let's solve each system of inequalities step-by-step and explain how to graph the solution for each. Since you can't see the image, I’ll walk through each problem with clear instructions on how to sketch the solution region on a coordinate plane.

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🔷 General Steps for Graphing Systems of Inequalities:

1. Graph each inequality as a line:
- Use solid line for ≤ or ≥.
- Use dashed line for < or >.
2. Shade the region that satisfies the inequality.
3. Find the overlapping shaded region — this is the solution to the system.

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## ✔ Problem 1:
$$
\begin{aligned}
(1)\quad & 3x + y \leq 2 \\
(2)\quad & 6x - 2y > -4
\end{aligned}
$$

Step 1: Rewrite in slope-intercept form (y = mx + b)

- (1): $ 3x + y \leq 2 \Rightarrow y \leq -3x + 2 $
- Solid line: $ y = -3x + 2 $
- Shade below the line.
- (2): $ 6x - 2y > -4 \Rightarrow -2y > -6x - 4 \Rightarrow y < 3x + 2 $
- Dashed line: $ y = 3x + 2 $
- Shade below the line.

Step 2: Find intersection point

Set $ -3x + 2 = 3x + 2 \Rightarrow -6x = 0 \Rightarrow x = 0 \Rightarrow y = 2 $

So they intersect at $ (0, 2) $

Step 3: Shade both regions and find overlap

- Both shaded areas are below their respective lines.
- The solution is where both conditions hold — the region below both lines, bounded by the two lines and the intersection.

✔ Final Answer: Shade the region below both lines, including the solid line $ y = -3x + 2 $, but not the dashed line $ y = 3x + 2 $.

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## ✔ Problem 2:
$$
\begin{aligned}
(1)\quad & 6x - 2y \leq 4 \\
(2)\quad & y < x
\end{aligned}
$$

Step 1: Rewrite

- (1): $ 6x - 2y \leq 4 \Rightarrow -2y \leq -6x + 4 \Rightarrow y \geq 3x - 2 $
- Solid line: $ y = 3x - 2 $
- Shade above the line.
- (2): $ y < x $
- Dashed line: $ y = x $
- Shade below the line.

Step 2: Find intersection

Set $ 3x - 2 = x \Rightarrow 2x = 2 \Rightarrow x = 1 \Rightarrow y = 1 $

Intersection: $ (1, 1) $

Step 3: Shade

- Above $ y = 3x - 2 $
- Below $ y = x $
- Overlap is the region between the two lines, above $ y = 3x - 2 $, below $ y = x $, and includes the solid line.

✔ Final Answer: Shade the region between the lines $ y = 3x - 2 $ (solid) and $ y = x $ (dashed), where $ y \geq 3x - 2 $ and $ y < x $. This forms a wedge-shaped region starting at $ (1,1) $.

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## ✔ Problem 3:
$$
\begin{aligned}
(1)\quad & 4x + y < 2 \\
(2)\quad & y > x
\end{aligned}
$$

Step 1: Rewrite

- (1): $ y < -4x + 2 $
- Dashed line: $ y = -4x + 2 $
- Shade below.
- (2): $ y > x $
- Dashed line: $ y = x $
- Shade above.

Step 2: Intersection

Set $ -4x + 2 = x \Rightarrow 2 = 5x \Rightarrow x = 0.4 \Rightarrow y = 0.4 $

Point: $ (0.4, 0.4) $

Step 3: Shade

- Region below $ y = -4x + 2 $
- Region above $ y = x $
- Overlap: between the lines, below the first, above the second.

✔ Final Answer: Shade the region between the two dashed lines, above $ y = x $ and below $ y = -4x + 2 $, forming a triangular-like area.

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## ✔ Problem 4:
$$
\begin{aligned}
(1)\quad & y < 2x + 1 \\
(2)\quad & y > -\frac{1}{3}x + 4
\end{aligned}
$$

Step 1: Lines

- (1): $ y = 2x + 1 $ → dashed line, shade below
- (2): $ y = -\frac{1}{3}x + 4 $ → dashed line, shade above

Step 2: Find intersection

Set $ 2x + 1 = -\frac{1}{3}x + 4 $

Multiply both sides by 3:
$ 6x + 3 = -x + 12 \Rightarrow 7x = 9 \Rightarrow x = \frac{9}{7} \approx 1.286 $

Then $ y = 2(\frac{9}{7}) + 1 = \frac{18}{7} + \frac{7}{7} = \frac{25}{7} \approx 3.57 $

Point: $ (\frac{9}{7}, \frac{25}{7}) $

Step 3: Shade

- Below $ y = 2x + 1 $
- Above $ y = -\frac{1}{3}x + 4 $
- Overlap: region between the two lines, bounded by their intersection.

✔ Final Answer: Shade the region between the two dashed lines, above the lower one and below the upper one.

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## ✔ Problem 5:
$$
\begin{aligned}
(1)\quad & 2x - y \geq -6 \\
(2)\quad & x > 2
\end{aligned}
$$

Step 1: Rewrite

- (1): $ 2x - y \geq -6 \Rightarrow -y \geq -2x - 6 \Rightarrow y \leq 2x + 6 $
- Solid line: $ y = 2x + 6 $
- Shade below.
- (2): $ x > 2 $
- Vertical dashed line at $ x = 2 $
- Shade to the right of it.

Step 2: Combine

- Region: $ x > 2 $ and $ y \leq 2x + 6 $
- This is the area to the right of $ x = 2 $, under the line $ y = 2x + 6 $

✔ Final Answer: Shade the region to the right of $ x = 2 $ and below $ y = 2x + 6 $. Include the line $ y = 2x + 6 $, exclude $ x = 2 $.

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## ✔ Problem 6:
$$
\begin{aligned}
(1)\quad & y < \frac{1}{2}x + 4 \\
(2)\quad & x - 2y \geq 6
\end{aligned}
$$

Step 1: Rewrite

- (1): $ y < \frac{1}{2}x + 4 $ → dashed line, shade below
- (2): $ x - 2y \geq 6 \Rightarrow -2y \geq -x + 6 \Rightarrow y \leq \frac{1}{2}x - 3 $
- Solid line: $ y = \frac{1}{2}x - 3 $
- Shade below

Wait! Both inequalities have same slope $ \frac{1}{2} $, but different intercepts.

- Line 1: $ y = \frac{1}{2}x + 4 $
- Line 2: $ y = \frac{1}{2}x - 3 $

Both shaded below, so we want region below both lines.

But since $ \frac{1}{2}x - 3 < \frac{1}{2}x + 4 $, the stricter condition is $ y \leq \frac{1}{2}x - 3 $, but wait:

Actually:
- $ y < \frac{1}{2}x + 4 $ → all points below the higher line
- $ y \leq \frac{1}{2}x - 3 $ → all points below or on the lower line

So the intersection is the region below or on $ y = \frac{1}{2}x - 3 $ (since that’s stricter).

But note: $ y \leq \frac{1}{2}x - 3 $ implies $ y < \frac{1}{2}x + 4 $ automatically (because $ \frac{1}{2}x - 3 < \frac{1}{2}x + 4 $)

So the solution is just $ y \leq \frac{1}{2}x - 3 $

✔ Final Answer: Graph $ y = \frac{1}{2}x - 3 $ as a solid line, shade below it. The other inequality is automatically satisfied.

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📌 Summary of Graphing Instructions:

| Problem | Lines | Shading | Notes |
|--------|-------|--------|-------|
| 1 | $ y = -3x+2 $ (solid), $ y=3x+2 $ (dashed) | Below both | Overlap region |
| 2 | $ y = 3x-2 $ (solid), $ y=x $ (dashed) | Above first, below second | Wedge between |
| 3 | $ y = -4x+2 $ (dashed), $ y=x $ (dashed) | Below first, above second | Between lines |
| 4 | $ y = 2x+1 $ (dashed), $ y = -\frac{1}{3}x+4 $ (dashed) | Below first, above second | Between lines |
| 5 | $ y = 2x+6 $ (solid), $ x=2 $ (dashed) | Below first, right of second | Right side under line |
| 6 | $ y = \frac{1}{2}x+4 $ (dashed), $ y = \frac{1}{2}x-3 $ (solid) | Below both → only need below $ y = \frac{1}{2}x - 3 $ | Parallel lines; tighter constraint wins |

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✔ Final Tip:

When graphing:
- Always use graph paper.
- Label axes.
- Draw lines carefully.
- Use solid/dashed correctly.
- Shade lightly and indicate the final overlapping region.

If you'd like, I can generate a visual description or help you sketch one specific problem in detail. Let me know!