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How to graph linear inequalities in two variables

Boundary first, shading second

Replace the inequality sign with an equals sign to find the boundary line. Draw a solid line for $\leq$ or $\geq$ and a dashed line for $<$ or $>$ . If the inequality is written as $y > m x + b$ , shade above; for $y < m x + b$ , shade below. Equality includes the boundary but does not change which side is shaded. Reverse the inequality sign whenever you multiply or divide both sides by a negative number.

Problem

Graph each inequality on a coordinate plane. State the boundary line, whether it is solid or dashed, and which side to shade.

1. $y \geq - \frac{1}{3} x - 6$ .

2. $y < x - 5$ .

3. $2 x - 5 y \leq - 10$ .

4. $y \leq - x - 2$ .

5. $y > - 2 x - 8$ .

6. $y \geq \frac{4}{3} x - 8$ .

7. $x - y < - 1$ .

8. $x - y < - 5$ .

9. $- 2 x + 5 y < 25$ .

10. $x - 2 y < - 4$ .

Example 2: the dashed boundary is y = x - 5. Only points below it satisfy the strict inequality; the line itself is excluded.

Answer

1. Solid, above. 2. Dashed, below. 3. Solid, above

y = \frac{2}{5} x + 2

. 4. Solid, below. 5. Dashed, above. 6. Solid, above. 7. Dashed, above

y = x + 1

. 8. Dashed, above

y = x + 5

. 9. Dashed, below

y = \frac{2}{5} x + 5

. 10. Dashed, above

y = \frac{1}{2} x + 2

Above and below mean vertically above or below the boundary at the same x-coordinate, not left or right of an axis.

Step-by-step solution

1. An inclusive inequality with negative slope

For $y \geq - \frac{1}{3} x - 6$ , the boundary is $y = - \frac{1}{3} x - 6$ . Two convenient points are $(0, - 6)$ and $(- 3, - 5)$ . Use a solid line and shade above. The origin passes the check $0 \geq - 6$ .

2. A strict inequality with positive slope

For $y < x - 5$ , draw the boundary through $(0, - 5)$ and $(5, 0)$ . Use a dashed line and shade below it. At the origin, $0 < - 5$ is false, so the origin must not be shaded. The diagram shows this case.

3. Divide by a negative coefficient carefully

Start with $2 x - 5 y \leq - 10$ . Subtract $2 x$ , then divide by $- 5$ and reverse the sign:

- 5 y \leq - 2 x - 10 ⟹ y \geq \frac{2}{5} x + 2.

The boundary passes through $(- 5, 0)$ and $(0, 2)$ . Draw it solid and shade above. The origin fails the original check $0 \leq - 10$ .

4. An inclusive inequality shaded below

For $y \leq - x - 2$ , use the boundary points $(- 2, 0)$ and $(0, - 2)$ . Draw a solid line and shade below. The origin fails $0 \leq - 2$ .

5. A strict inequality with a steeper negative slope

For $y > - 2 x - 8$ , the boundary passes through $(- 4, 0)$ and $(0, - 8)$ . Draw it dashed and shade above. The origin satisfies $0 > - 8$ .

6. A positive fractional slope

For $y \geq \frac{4}{3} x - 8$ , plot $(0, - 8)$ and $(6, 0)$ . The line is solid, and the region above it is shaded. The origin satisfies $0 \geq - 8$ .

7. Rearrange x minus y

From $x - y < - 1$ , subtract $x$ and multiply by $- 1$ :

- y < - x - 1 ⟹ y > x + 1.

Use a dashed line through $(- 1, 0)$ and $(0, 1)$ , with shading above. The origin is not a solution.

8. The same slope with a different intercept

Similarly, $x - y < - 5$ becomes $y > x + 5$ . Draw a dashed line through $(- 5, 0)$ and $(0, 5)$ and shade above. Again, the origin is outside the solution region.

9. Choose a visible second point

For $- 2 x + 5 y < 25$ , solve for $y$ :

5 y < 2 x + 25 ⟹ y < \frac{2}{5} x + 5.

Draw a dashed line through $(0, 5)$ and $(5, 7)$ , then shade below. The x-intercept is $(- 12.5, 0)$ ; if it is off the grid, the other two points are enough. The origin satisfies $0 < 25$ .

10. Another negative divisor

For $x - 2 y < - 4$ , subtract $x$ and divide by $- 2$ :

- 2 y < - x - 4 ⟹ y > \frac{1}{2} x + 2.

Draw a dashed line through $(0, 2)$ and $(2, 3)$ and shade above. The origin fails $0 < - 4$ .

Check your graph

Test a point that is not on the boundary in the original inequality. If it satisfies the inequality, shade the half-plane containing that point. If it does not, shade the other half-plane. When the origin lies on the boundary, choose another test point.

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