Two Variable System of Inequalities - Graphing worksheet with seven problems showing linear inequalities and their solutions on graphs.

Let’s solve each problem step by step. Each one shows a graph of two inequalities, and we need to find the solution region — that’s where both shaded areas overlap.

We’ll go one by one.

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Problem 1:

Inequalities:
- y ≥ -½x + 4 → solid line, shade above
- y < 2x - 1 → dashed line, shade below

Look at the graph: The overlapping region is the pink area in the bottom right. That matches point (8,0) — let’s check if it satisfies both:

Check (8,0):
- First inequality: 0 ≥ -½(8) + 4 → 0 ≥ -4 + 4 → 0 ≥ 0 ✔
- Second inequality: 0 < 2(8) - 1 → 0 < 16 - 1 → 0 < 15 ✔

So (8,0) works.

✔ Answer for #1: (8,0)

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Problem 2:

Inequalities:
- y ≤ -⅓x + 10 → solid line, shade below
- y > 3x - 10 → dashed line, shade above

Graph shows green shaded region near top left. Try point (2,9):

Check (2,9):
- First: 9 ≤ -⅓(2) + 10 → 9 ≤ -0.67 + 10 → 9 ≤ 9.33 ✔
- Second: 9 > 3(2) - 10 → 9 > 6 - 10 → 9 > -4 ✔

Works!

✔ Answer for #2: (2,9)

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Problem 3:

Inequalities:
- y > 2x - 3 → dashed, shade above
- x ≤ 2 → solid vertical line, shade left

Shaded region is red on the left side, up to x=2. Try point (-2,0):

Check (-2,0):
- First: 0 > 2(-2) - 3 → 0 > -4 - 3 → 0 > -7 ✔
- Second: -2 ≤ 2 ✔

Perfect.

✔ Answer for #3: (-2,0)

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Problem 4:

Inequalities:
- y ≥ ½x + 2 → solid, shade above
- x + 3y ≥ 6 → rewrite as y ≥ -⅓x + 2 → solid, shade above

Green shaded region is small triangle near origin? Wait — look at graph: shaded region is between lines, starting around (0,2). Try point (0,2):

Check (0,2):
- First: 2 ≥ ½(0) + 2 → 2 ≥ 2 ✔
- Second: 0 + 3(2) = 6 ≥ 6 ✔

Yes! Also, (0,2) is labeled on the graph.

✔ Answer for #4: (0,2)

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Problem 5:

Inequalities:
- y ≤ ¾x + 3 → solid, shade below
- y ≥ 2x - 3 → solid, shade above

Pink shaded region is in middle. Try point (-4,-2):

Check (-4,-2):
- First: -2 ≤ ¾(-4) + 3 → -2 ≤ -3 + 3 → -2 ≤ 0 ✔
- Second: -2 ≥ 2(-4) - 3 → -2 ≥ -8 - 3 → -2 ≥ -11 ✔

Good.

✔ Answer for #5: (-4,-2)

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Problem 6:

Inequalities:
- 2x + 7y ≥ 0 → rewrite: y ≥ -²/₇x → solid, shade above
- 3x - 2y ≤ 12 → rewrite: y ≥ ³/₂x - 6 → solid, shade above? Wait — let’s rearrange properly:

3x - 2y ≤ 12
→ -2y ≤ -3x + 12
→ y ≥ ³/₂x - 6 (flip sign when dividing by negative)

Wait — actually, let’s test point (3,0) from graph label.

Check (3,0):
First inequality: 2(3) + 7(0) = 6 ≥ 0 ✔
Second: 3(3) - 2(0) = 9 ≤ 12 ✔

And graph shows green shading including (3,0).

✔ Answer for #6: (3,0)

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Problem 7:

Inequalities:
- y ≥ -⅔x + 4 → solid, shade above
- x + 6y ≤ 18 → rewrite: y ≤ -¹/₆x + 3 → solid, shade below

Red shaded region is small triangle near bottom left. Try point (6,0):

Check (6,0):
First: 0 ≥ -⅔(6) + 4 → 0 ≥ -4 + 4 → 0 ≥ 0 ✔
Second: 6 + 6(0) = 6 ≤ 18 ✔

Also labeled on graph.

✔ Answer for #7: (6,0)

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Problem 8:

Inequalities:
- 3x + 7y ≤ 52 → rewrite: y ≤ -³/₇x + 52/7 ≈ -0.43x + 7.43 → solid, shade below
- 2x - 5y ≥ 15 → rewrite: -5y ≥ -2x + 15 → y ≤ ²/₅x - 3 → solid, shade below

Wait — let’s test point (5,-1) from graph label.

Check (5,-1):
First: 3(5) + 7(-1) = 15 - 7 = 8 ≤ 52 ✔
Second: 2(5) - 5(-1) = 10 + 5 = 15 ≥ 15 ✔

Perfect.

✔ Answer for #8: (5,-1)

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Final Answer:
1) (8,0)
2) (2,9)
3) (-2,0)
4) (0,2)
5) (-4,-2)
6) (3,0)
7) (6,0)
8) (5,-1)