Linear Inequalities worksheet with six equations to graph and shade on coordinate grids.

Let’s solve each inequality one by one. We’ll graph the line first, then decide which side to shade based on the inequality sign.

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Problem 1: y ≤ -⅓x + 2

Step 1: Graph the line y = -⅓x + 2
- When x = 0, y = 2 → point (0, 2)
- When x = 3, y = -1 + 2 = 1 → point (3, 1)
- When x = -3, y = 1 + 2 = 3 → point (-3, 3)

Draw a solid line (because of “≤”) through these points.

Step 2: Shade below the line (since y is less than or equal to).

✔ Final for #1: Solid line with slope -⅓, y-intercept 2; shade below.

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Problem 2: y ≤ ½x - 5

Step 1: Graph y = ½x - 5
- When x = 0, y = -5 → (0, -5)
- When x = 2, y = 1 - 5 = -4 → (2, -4)
- When x = 10, y = 5 - 5 = 0 → (10, 0)

Solid line (≤), shade below.

✔ Final for #2: Solid line, slope ½, y-intercept -5; shade below.

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Problem 3: y ≥ ⅛x + 2

Step 1: Graph y = ⅛x + 2
- When x = 0, y = 2 → (0, 2)
- When x = 8, y = 1 + 2 = 3 → (8, 3)
- When x = -8, y = -1 + 2 = 1 → (-8, 1)

Solid line (≥), shade above.

✔ Final for #3: Solid line, slope ⅛, y-intercept 2; shade above.

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Problem 4: y > ¼x - 3

Step 1: Graph y = ¼x - 3
- When x = 0, y = -3 → (0, -3)
- When x = 4, y = 1 - 3 = -2 → (4, -2)
- When x = 8, y = 2 - 3 = -1 → (8, -1)

Dashed line (because “>”, not “≥”), shade above.

✔ Final for #4: Dashed line, slope ¼, y-intercept -3; shade above.

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Problem 5: ⅙x + ¹⁄₁₀y ≤ ½

We need to rewrite this in slope-intercept form (y = mx + b).

Multiply entire inequality by 30 (LCM of 6, 10, 2) to eliminate denominators:

30*(⅙x) + 30*(¹⁄₁₀y) ≤ 30*(½)
→ 5x + 3y ≤ 15

Now solve for y:

3y ≤ -5x + 15
y ≤ -⁵⁄x + 5

Graph y = -⁵⁄₃x + 5
- When x = 0, y = 5 → (0, 5)
- When x = 3, y = -5 + 5 = 0 → (3, 0)
- When x = -3, y = 5 + 5 = 10 → (-3, 10) — but that’s off grid, so use (3,0) and (0,5)

Solid line (≤), shade below.

✔ Final for #5: Solid line, slope -⁵⁄, y-intercept 5; shade below.

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Problem 6: 5x ≤ -4y - 12

Solve for y:

Add 4y to both sides:
5x + 4y ≤ -12

Subtract 5x:
4y ≤ -5x - 12

Divide by 4:
y ≤ -⁵⁄₄x - 3

Graph y = -⁵⁄₄x - 3
- When x = 0, y = -3 → (0, -3)
- When x = 4, y = -5 - 3 = -8 → (4, -8)
- When x = -4, y = 5 - 3 = 2 → (-4, 2)

Solid line (≤), shade below.

✔ Final for #6: Solid line, slope -⁵⁄₄, y-intercept -3; shade below.

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Final Answer:
For each problem, graph the corresponding line (solid if ≤ or ≥, dashed if < or >), then shade the region that satisfies the inequality:

1. Line: y = -⅓x + 2 (solid), shade below
2. Line: y = ½x - 5 (solid), shade below
3. Line: y = ⅛x + 2 (solid), shade above
4. Line: y = ¼x - 3 (dashed), shade above
5. Line: y = -⁵⁄₃x + 5 (solid), shade below
6. Line: y = -⁵⁄₄x - 3 (solid), shade below