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Problem 1: Graph $4x - 6y \leq -12$.
- Solve for $y$: $-6y \leq -4x - 12$, so $y \geq \frac{2}{3}x + 2$.
- Graph the line $y = \frac{2}{3}x + 2$ with a solid line (since $\geq$).
- Shade above the line (since $y \geq$).
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Problem 2: Write the slope-intercept form of the graphed inequality.
- The line passes through $(0, 0)$ and has slope $-1$, so equation is $y = -x$.
- The shaded region is below the dashed line, so inequality is $y < -x$.
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Problem 3: Which inequality best represents the graph? Circle your answer.
- The line passes through $(0, -4)$ and $(4, 0)$, so slope is $1$, y-intercept is $-4$, equation is $y = x - 4$.
- Shaded region is above the solid line, so $y \geq x - 4$.
- Convert to standard form: $x - y \leq 4$. Multiply by 5: $5x - 5y \leq 20$ — not an option.
- Check options:
- a. $10x - 5y \geq 20$ → $y \leq 2x - 4$ — wrong direction.
- b. $5x - 10y \leq 20$ → $y \geq \frac{1}{2}x - 2$ — wrong slope.
- c. $10x - 5y > 20$ → $y < 2x - 4$ — wrong direction and strict.
- d. $5x - 10y < 20$ → $y > \frac{1}{2}x - 2$ — wrong slope.
- None match exactly. Recheck: Line from (-4,0) to (0,-4)? Slope = (-4-0)/(0-(-4)) = -1, y-int = -4 → $y = -x -4$.
- Shaded above → $y \geq -x -4$ → $x + y \geq -4$.
- Multiply by 5: $5x + 5y \geq -20$ — not listed.
- Perhaps line is $y = \frac{1}{2}x - 2$? From (0,-2) to (4,0), slope=0.5.
- Shaded above → $y \geq \frac{1}{2}x - 2$ → multiply by 10: $10y \geq 5x - 20$ → $-5x + 10y \geq -20$ → $5x - 10y \leq 20$ — matches option b.
- So answer is
b.
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Problem 4: Graph $y > \frac{5}{2}x - 3$.
- Graph line $y = \frac{5}{2}x - 3$ with dashed line (since >).
- Shade above the line.
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Problem 5: Graph $-8x + 9y \geq 72$.
- Solve for $y$: $9y \geq 8x + 72$, so $y \geq \frac{8}{9}x + 8$.
- Graph line $y = \frac{8}{9}x + 8$ with solid line.
- Shade above.
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Problem 6: Write the slope-intercept form of the graphed inequality.
- Line passes through (0, -1) and (2, 1), slope = (1 - (-1))/(2 - 0) = 1, y-int = -1 → $y = x - 1$.
- Shaded region is above the dashed line → $y > x - 1$.
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Problem 7: Graph $y < 3$.
- Draw horizontal line at $y = 3$ with dashed line.
- Shade below the line.
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Problem 8: Which inequality best represents the graph? Circle your answer.
- Vertical line at $x = -1$, shaded to the right.
- Solid line → includes equality → $x \geq -1$.
- Answer is
a.
Parent Tip: Review the logic above to help your child master the concept of graph linear inequalities worksheet.
