Graphing Linear Inequalities in Two Variables worksheet | Live ... - Free Printable

Let’s go step by step to solve each part of the worksheet.

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Part I: Review Questions

1. What is the graph of a linear inequality in two variables?
→ It’s a region on the coordinate plane (a shaded area) that shows all the solutions to the inequality. The boundary line separates the solution region from the non-solution region.

2. What is the boundary line when graphing a linear inequality that uses > or <?
→ It’s a dashed line, because the points ON the line are NOT included in the solution.

3. What is the boundary line when graphing a linear inequality that uses ≥ or ≤?
→ It’s a solid line, because the points ON the line ARE included in the solution.

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Part II: Match the correct symbol to each graph

We look at:
- Whether the line is solid or dashed → tells us if it’s ≥/≤ or >/ <
- Which side is shaded → test a point not on the line (like (0,0)) to see which inequality fits.

Problem 1: y ___ -9x + 5
Graph has a dashed vertical line? Wait — actually, looking again: the graph shows a vertical dashed line at x = 1? But the equation is written as “y ___ -9x + 5” — that doesn’t match a vertical line. Let me recheck.

Wait — actually, looking at Graph 1: it’s a vertical dashed line at x = 1, and the shading is to the LEFT. But the equation given is “y ___ -9x + 5” — that must be a typo or mislabel? Actually, no — let’s look carefully.

Actually, Graph 1: The line is vertical, so it should be an inequality like x < 1 or x > 1. But the problem says “y ___ -9x + 5”. That suggests maybe the graph is mislabeled? Or perhaps I’m misreading.

Wait — let’s look at the actual graphs:

Graph 1: Vertical dashed line at x=1, shaded left → so inequality is x < 1. But the problem writes “y ___ -9x + 5” — that can’t be right for a vertical line. Maybe it’s a mistake in the worksheet? Or perhaps I need to interpret differently.

Actually, let’s check the other graphs first.

Graph 2: Line with positive slope, solid line, shaded below → so y ≤ 2x - 3? Let’s test (0,0): 0 ≤ 0 - 3? 0 ≤ -3? False. So if (0,0) is not shaded, but in this graph, (0,0) is above the line? Wait, the line passes through (0,-3) and (1.5,0). Shading is BELOW the line. Test (0,0): plug into y ? 2x - 3 → 0 ? 0 - 3 → 0 ? -3. Since 0 > -3, and (0,0) is NOT in the shaded region, then the inequality must be y ≤ 2x - 3? But 0 ≤ -3 is false, so (0,0) is not a solution — which matches since it’s not shaded. So yes, y ≤ 2x - 3.

But wait — the line is SOLID, so it must be ≤ or ≥. And since shading is below, it’s ≤.

So Problem 2: ≤

Graph 3: Line with negative slope, solid line, shaded below. Equation: x + y ___ 0. Test (0,0): 0+0=0, and (0,0) is on the line. Now test a point below, say (0,-1): 0 + (-1) = -1 < 0. Is (0,-1) shaded? Yes. So x + y ≤ 0. Because -1 ≤ 0 is true. So ≤

Graph 4: Dashed line, positive slope, shaded above. Equation: 5x - 4y ___ -12. Test (0,0): 0 - 0 = 0 ___ -12. 0 > -12. Is (0,0) shaded? In the graph, (0,0) is below the line, and shading is ABOVE the line. So (0,0) is NOT shaded. So 0 is not greater than -12 in the solution set? Wait, if shading is above, and (0,0) is below and not shaded, then the inequality should be such that (0,0) does NOT satisfy it.

Plug (0,0) into 5x - 4y > -12 → 0 > -12 → true. But (0,0) is not in the shaded region, so it should NOT satisfy the inequality. Contradiction.

Try 5x - 4y < -12 → 0 < -12? False. So (0,0) does not satisfy it — good, since it’s not shaded. But is the shading above or below?

The line: 5x - 4y = -12. When x=0, -4y=-12 → y=3. When y=0, 5x=-12 → x=-2.4. So line goes from (-2.4,0) to (0,3). Slope is positive. Shading is ABOVE the line? In the graph, the shaded region is above the dashed line. For example, point (0,4): 5(0)-4(4)= -16. Is -16 < -12? Yes. And (0,4) is above the line and shaded. Point (0,0): 0 > -12, but not shaded. So if we want (0,4) to satisfy and (0,0) not to, then 5x - 4y < -12 would work for (0,4): -16 < -12 true; for (0,0): 0 < -12 false — perfect. But the line is dashed, so strict inequality. So <

But let's confirm direction. If we solve for y: 5x - 4y < -12 → -4y < -5x -12 → divide by -4 (reverse inequality): y > (5/4)x + 3. So y > ... means shading ABOVE the line. Yes! So the inequality is 5x - 4y < -12, which is equivalent to y > (5/4)x + 3. So symbol is <

Now back to Problem 1: y ___ -9x + 5

Graph 1: It's a vertical dashed line at x=1, shaded to the left. But the equation is written in terms of y. This must be an error. Perhaps it's supposed to be x ___ something? Or maybe the graph is for a different equation.

Looking again: the graph has a vertical line, so it can't be y = -9x + 5, which is slanted. There's a mismatch.

Perhaps it's a typo, and it's meant to be x ___ 1 or something. But the problem says "y ___ -9x + 5".

Another possibility: maybe the graph is not for that equation? But the worksheet pairs them.

Let me assume that for Graph 1, even though it looks vertical, perhaps it's very steep? -9x + 5 has slope -9, which is very steep downward. At x=0, y=5; x=1, y=-4. So from (0,5) to (1,-4), that's a steep line. In the graph, it might be drawn as almost vertical.

In Graph 1, the line is dashed, and shaded to the left. For a line with negative slope, "left" might mean where x is smaller.

Test a point. Say, for y ? -9x + 5. Take a point in the shaded region. From the graph, say (-1,0): is it shaded? Assuming yes. Plug in: y = 0, -9x +5 = -9(-1)+5=9+5=14. So 0 ? 14. If shaded region satisfies the inequality, and 0 < 14, so if it's y < -9x +5, then 0<14 true. Now take a point not shaded, say (2,0): -9(2)+5=-18+5=-13. 0 ? -13. If y < -9x+5, then 0 < -13? False, and (2,0) is not shaded — good. Also, the line is dashed, so strict inequality. So <

Yes, that makes sense. Even though the line is steep, it's not vertical. So for Problem 1: <

Summary for Part II:

1. y < -9x + 5
2. y ≤ 2x - 3
3. x + y ≤ 0
4. 5x - 4y < -12

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Part III: Check if points are solutions to x - y > -2

The inequality is x - y > -2. We plug in each point and see if the inequality holds.

Also, note the graph: dashed line, shaded region. The line is x - y = -2, or y = x + 2. Dashed, so not including line. Shaded region: for example, origin (0,0): 0-0=0 > -2? True, and (0,0) is shaded? In the graph, the pink region includes (0,0)? Looking at the grid, the line goes through (-2,0) and (0,2), and shading is below the line? Let's see.

Point A (-3,1) is given as NO. Check: x-y = -3 -1 = -4 > -2? -4 > -2? False, so NO — correct.

Now others:

B (2, -1): x-y = 2 - (-1) = 2+1=3 > -2? True → YES

C (1,2): 1-2= -1 > -2? True → YES

D (-1,1): -1 -1 = -2 > -2? -2 > -2 is false (since not equal) → NO

E (-2,2): -2 -2 = -4 > -2? False → NO

But let's confirm with the graph. The shaded region is where x - y > -2. For point B (2,-1): should be in shaded region. Similarly, C (1,2): 1-2=-1>-2, yes. D (-1,1): -2 not greater than -2, so on the line? But line is dashed, so not included. E (-2,2): -4 < -2, not satisfied.

So answers:

A: NO (given)
B: YES
C: YES
D: NO
E: NO

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Final Answers:

Part I:
1. A shaded region on the coordinate plane representing all solutions.
2. A dashed line.
3. A solid line.

Part II:
1. <
2. ≤
3. ≤
4. <

Part III:
B: YES
C: YES
D: NO
E: NO

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Final Answer:
Part I:
1. A shaded region on the coordinate plane
2. Dashed line
3. Solid line

Part II:
1. <
2. ≤
3. ≤
4. <

Part III:
B: YES
C: YES
D: NO
E: NO