Writing Linear Inequalities Worksheet featuring six graphs to derive inequalities from given shaded regions.

Let's solve each of the six problems on the "Writing Linear Inequalities Worksheet" by analyzing the graphs and deriving the correct inequality for each. We’ll go step-by-step.

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General Steps to Write an Inequality from a Graph:

1. Identify the line: Find the equation of the boundary line (slope-intercept form: $ y = mx + b $).
2. Determine if the line is solid or dashed:
- Solid line → includes equality (≤ or ≥)
- Dashed line → strict inequality (< or >)
3. Determine which side is shaded:
- Pick a test point (like the origin (0,0)) and see if it satisfies the inequality.
- If yes, that side is shaded; otherwise, the other side is.
4. Write the inequality using the line equation and appropriate inequality sign.

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Let’s go through each graph:

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Problem 1

- Line: Passes through (0, 2) and (6, -4).
Slope $ m = \frac{-4 - 2}{6 - 0} = \frac{-6}{6} = -1 $
So, $ y = -x + 2 $
- Line is solid → includes equality
- Shaded region: Below the line (includes points like (0,0))
- Test (0,0): $ 0 \leq -0 + 2 $ → $ 0 \leq 2 $ → True
- Inequality: $ y \leq -x + 2 $

✔ Answer: $ y \leq -x + 2 $

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Problem 2

- Line: Passes through (0, 2) and (2, 8)
Slope $ m = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3 $
Equation: $ y = 3x + 2 $
- Line is solid
- Shaded region: Above the line? Let’s test (0,0):
$ 0 \geq 3(0) + 2 $? → $ 0 \geq 2 $? No → so shaded region is not above
Try a point in shaded area, e.g., (0, -5): $ -5 \geq 2 $? No → wait, look at graph: shaded is below the line?
Wait — actually, shaded region is to the left of the line? No — let’s recheck.

Wait: The shaded region is above the line? But the line goes up from bottom-left to top-right. Shaded region is on the right side of the line?

Wait — no: look carefully. The shaded region is to the left of the line? Actually, no.

Let me analyze:

The line passes through (0,2), (2,8), and (−2, −4). It has positive slope.

But the shaded region is below the line? Let’s test (0,0):

Is (0,0) shaded? Yes — it's in the blue area.

Check: $ 0 \leq 3(0) + 2 $ → $ 0 \leq 2 $ → True

So shaded region is below the line → $ y \leq 3x + 2 $

But wait — is the line solid? Yes.

And shaded region includes (0,0), which is below the line.

✔ Answer: $ y \leq 3x + 2 $

Wait — but looking at the graph again: the shaded region appears to be on the side where x is small, but since the line has positive slope, and shaded is below, yes.

But wait — let’s check another point: say (0, 10) — not shaded. (0, 0) is shaded. (0, 3): $ 3 \leq 2 $? No → so only below.

Yes, so shaded region is below the line.

✔ Answer: $ y \leq 3x + 2 $

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Problem 3

- Vertical line at $ x = 2 $
- Solid line → includes equality
- Shaded region: To the right of $ x = 2 $
- Test point: (3,0) → $ x = 3 > 2 $ → shaded
- (1,0) → not shaded
- So: $ x \geq 2 $

✔ Answer: $ x \geq 2 $

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Problem 4

- Line: Passes through (0, 9) and (6, -6)
- Slope $ m = \frac{-6 - 9}{6 - 0} = \frac{-15}{6} = -\frac{5}{2} $
- Equation: $ y = -\frac{5}{2}x + 9 $
- Solid line → includes equality
- Shaded region: Below the line? Test (0,0):
$ 0 \leq -\frac{5}{2}(0) + 9 $? → $ 0 \leq 9 $ → True → so shaded region includes origin
- So: $ y \leq -\frac{5}{2}x + 9 $

✔ Answer: $ y \leq -\frac{5}{2}x + 9 $

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Problem 5

- Line: Passes through (0, -2) and (4, 2)
- Slope $ m = \frac{2 - (-2)}{4 - 0} = \frac{4}{4} = 1 $
- Equation: $ y = x - 2 $
- Dashed line → strict inequality
- Shaded region: Below the line? Test (0,0):
$ 0 < 0 - 2 $? → $ 0 < -2 $? False → so not shaded
Try (0, -3): $ -3 < 0 - 2 $? → $ -3 < -2 $? True → and (0,-3) is shaded? Yes — looks like shaded region is below the line
But (0,0) is not shaded, and (0,-3) is shaded → so shaded region is below the line

Wait: $ y < x - 2 $? At (0,-3): $ -3 < -2 $? Yes → true

But is the line dashed? Yes → so strict inequality

So: $ y < x - 2 $

Wait — but look: the shaded region is below the line, and line is dashed → yes.

✔ Answer: $ y < x - 2 $

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Problem 6

- Same line as Problem 5: $ y = x - 2 $
- But line is solid → includes equality
- Shaded region: Same as before? Still below? Let’s test (0,0):
$ 0 \leq 0 - 2 $? → $ 0 \leq -2 $? False → so (0,0) is not shaded
Try (0,-3): $ -3 \leq -2 $? True → and it is shaded

So shaded region is below the line → $ y \leq x - 2 $

✔ Answer: $ y \leq x - 2 $

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✔ Final Answers:

1. $ y \leq -x + 2 $
2. $ y \leq 3x + 2 $
3. $ x \geq 2 $
4. $ y \leq -\frac{5}{2}x + 9 $
5. $ y < x - 2 $
6. $ y \leq x - 2 $

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