Printable worksheet for practicing graphing linear inequalities, featuring nine equations to be graphed on coordinate planes.

Of course! I'll solve each of the 9 linear inequalities step-by-step and explain how to graph them. Since I can't draw on the image, I’ll describe exactly what you need to sketch on each grid.

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General Rules for Graphing Linear Inequalities:

1. Graph the boundary line: Treat the inequality as an equation (e.g., `y = ...`). If the inequality is `≤` or `≥`, draw a solid line (includes the line). If it’s `<` or `>`, draw a dashed line (excludes the line).

2. Shade the correct region: Pick a test point not on the line (usually `(0,0)` if it’s not on the line) and plug it into the inequality. If the inequality is true, shade the side containing that point. If false, shade the opposite side.

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1) y ≥ 2

- Boundary line: `y = 2` → horizontal line at y=2.
- Solid line because of `≥`.
- Test point (0,0): Is `0 ≥ 2`? No → shade above the line.
- ✔ Sketch: Draw solid horizontal line at y=2. Shade everything above it.

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2) y < 3x - 1

- Boundary line: `y = 3x - 1`
- Slope = 3, y-intercept = -1 → goes through (0,-1), then up 3, right 1 → (1,2)
- Dashed line because of `<`.
- Test point (0,0): Is `0 < 3(0) - 1`? → `0 < -1`? No → shade below the line.
- ✔ Sketch: Draw dashed line through (0,-1) and (1,2). Shade below it.

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3) y ≤ -7x + 3

- Boundary line: `y = -7x + 3`
- Slope = -7, y-intercept = 3 → goes through (0,3), then down 7, right 1 → (1,-4)
- Solid line because of `≤`.
- Test point (0,0): Is `0 ≤ -7(0) + 3`? → `0 ≤ 3`? Yes → shade below the line.
- ✔ Sketch: Draw solid line through (0,3) and (1,-4). Shade below it.

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4) y > 8x

- Boundary line: `y = 8x`
- Slope = 8, y-intercept = 0 → goes through (0,0), then up 8, right 1 → (1,8)
- Dashed line because of `>`.
- Test point (0,1): Is `1 > 8(0)`? → `1 > 0`? Yes → shade above the line.
*(Note: Can’t use (0,0) since it’s ON the line.)*
- ✔ Sketch: Draw dashed line through (0,0) and (1,8). Shade above it.

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5) y ≥ -2 - 5x → same as y ≥ -5x - 2

- Boundary line: `y = -5x - 2`
- Slope = -5, y-intercept = -2 → goes through (0,-2), then down 5, right 1 → (1,-7)
- Solid line because of `≥`.
- Test point (0,0): Is `0 ≥ -5(0) - 2`? → `0 ≥ -2`? Yes → shade above the line.
- ✔ Sketch: Draw solid line through (0,-2) and (1,-7). Shade above it.

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6) y ≤ (3/2)(x + 1)

First, simplify: `y ≤ (3/2)x + 3/2`

- Boundary line: `y = (3/2)x + 1.5`
- Slope = 3/2, y-intercept = 1.5 → goes through (0, 1.5), then up 3, right 2 → (2, 4.5)
- Solid line because of `≤`.
- Test point (0,0): Is `0 ≤ (3/2)(0+1)`? → `0 ≤ 1.5`? Yes → shade below the line.
- ✔ Sketch: Draw solid line through (0, 1.5) and (2, 4.5). Shade below it.

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7) y < -4 + x → same as y < x - 4

- Boundary line: `y = x - 4`
- Slope = 1, y-intercept = -4 → goes through (0,-4), then up 1, right 1 → (1,-3)
- Dashed line because of `<`.
- Test point (0,0): Is `0 < 0 - 4`? → `0 < -4`? No → shade below the line.
- ✔ Sketch: Draw dashed line through (0,-4) and (1,-3). Shade below it.

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8) y ≤ -3

- Boundary line: `y = -3` → horizontal line at y=-3.
- Solid line because of `≤`.
- Test point (0,0): Is `0 ≤ -3`? No → shade below the line.
- ✔ Sketch: Draw solid horizontal line at y=-3. Shade everything below it.

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9) y > x + 1

- Boundary line: `y = x + 1`
- Slope = 1, y-intercept = 1 → goes through (0,1), then up 1, right 1 → (1,2)
- Dashed line because of `>`.
- Test point (0,0): Is `0 > 0 + 1`? → `0 > 1`? No → shade above the line.
- ✔ Sketch: Draw dashed line through (0,1) and (1,2). Shade above it.

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## ✔ Final Answer Summary (What to Sketch):

| Problem | Line Type | Key Points | Shade Region |
|---------|-----------|------------|--------------|
| 1) y ≥ 2 | Solid horizontal at y=2 | — | Above |
| 2) y < 3x - 1 | Dashed, slope 3, y-int -1 | (0,-1), (1,2) | Below |
| 3) y ≤ -7x + 3 | Solid, slope -7, y-int 3 | (0,3), (1,-4) | Below |
| 4) y > 8x | Dashed, slope 8, y-int 0 | (0,0), (1,8) | Above |
| 5) y ≥ -5x - 2 | Solid, slope -5, y-int -2 | (0,-2), (1,-7) | Above |
| 6) y ≤ (3/2)(x+1) | Solid, slope 3/2, y-int 1.5 | (0,1.5), (2,4.5) | Below |
| 7) y < x - 4 | Dashed, slope 1, y-int -4 | (0,-4), (1,-3) | Below |
| 8) y ≤ -3 | Solid horizontal at y=-3 | — | Below |
| 9) y > x + 1 | Dashed, slope 1, y-int 1 | (0,1), (1,2) | Above |

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You’re all set! Just follow these instructions on your worksheet grids. Let me know if you’d like printable versions or step-by-step graphs drawn for you.