Graphing Linear Inequalities in Two Variables
To graph a linear inequality in two variables, graph its boundary line, then shade the side that holds the solutions. Use a solid line for ≤ or ≥ and a dashed line for < or >. To choose a side, test a point such as (0, 0). If it makes the inequality true, shade the side with that point.
Before this lesson: You should know how to graph a line in slope-intercept form and how to solve multi-step inequalities, including when to flip the inequality sign.
What Does the Graph of an Inequality Show?
A linear equation like y = x + 1 has solutions that make a line. A linear inequality like y \gt x + 1 has many more solutions. They fill a whole region on one side of the line.
Try two points in y \gt x + 1:
- (0, 3): 3 \gt 0 + 1 is true. So (0, 3) is a solution.
- (0, 0): 0 \gt 0 + 1 is false. So (0, 0) is not a solution.
The line y = x + 1 is called the boundary line. It splits the grid into two halves. Every point on one side is a solution, and every point on the other side is not.
Solid or Dashed Line?
The inequality sign tells you whether points on the boundary line are solutions.
| Symbol | Meaning | Boundary line |
|---|---|---|
| \lt | less than | Dashed (points on the line don’t count) |
| \gt | greater than | Dashed (points on the line don’t count) |
| \le | less than or equal to | Solid (points on the line count) |
| \ge | greater than or equal to | Solid (points on the line count) |
How to Graph a Linear Inequality
- Get y alone if you can. If you multiply or divide by a negative number, flip the inequality sign.
- Graph the boundary line. Replace the inequality sign with = and graph that line. Make it dashed for < or >, and solid for ≤ or ≥.
- Test a point that is not on the line. (0, 0) is easiest unless the line goes through it.
- Shade. If the test point makes the inequality true, shade its side. If not, shade the other side.
Shortcut: When y is alone on the left, y \gt or y \ge means shade above the line. y \lt or y \le means shade below. Still test a point to check.
Worked Examples
Example 1: Check a point
Is (2, 1) a solution of y \lt 3x - 4?
Step 1. Substitute x = 2 and y = 1: 1 \lt 3(2) - 4.
Step 2. Simplify the right side: 1 \lt 2. This is true.
Answer: Yes, (2, 1) is a solution.
Example 2: Graph a “greater than or equal to” inequality
Graph y \ge -x + 3.
Step 1. y is already alone.
Step 2. Graph the boundary line y = -x + 3: start at (0, 3), then go down 1 and right 1. The sign is ≥, so draw a solid line.
Step 3. Test (0, 0): 0 \ge -0 + 3 gives 0 \ge 3, which is false.
Step 4. (0, 0) is below the line and is not a solution, so shade above the line.
Example 3: Flip the sign
Graph 2x - 3y \gt 6.
Step 1. Get y alone. Subtract 2x, then divide by −3. Dividing by a negative flips > to <.
\begin{aligned}
2x - 3y &> 6 \\
-3y &> -2x + 6 \\
y &< \tfrac{2}{3}x - 2
\end{aligned}Step 2. Graph the boundary line y = \frac{2}{3}x - 2: start at (0, −2), then go up 2 and right 3. The sign is <, so draw a dashed line.
Step 3. Test (0, 0) in the original inequality: 2(0) - 3(0) \gt 6 gives 0 \gt 6, which is false.
Step 4. (0, 0) is above the line, so shade below the line. This matches the < sign.
Example 4: Horizontal and vertical boundary lines
Graph x \lt 2 and, separately, y \ge -1.
Step 1. For x \lt 2, draw a dashed vertical line at x = 2. Points with x-values less than 2 are to the left, so shade left.
Step 2. For y \ge -1, draw a solid horizontal line at y = -1. Points with y-values of −1 or more are on or above it, so shade above.
Example 5: Word problem
A school club sells student tickets for $5 and adult tickets for $10. It needs to raise at least $200. Write and graph an inequality. Will 20 student tickets and 12 adult tickets be enough?
Step 1. Let s be the number of student tickets and a the number of adult tickets. “At least” means ≥.
5s + 10a \ge 200
Step 2. Graph the boundary line 5s + 10a = 200 using its intercepts. If a = 0, then s = 40. If s = 0, then a = 20. Draw a solid line through (40, 0) and (0, 20).
Step 3. Test (0, 0): 0 \ge 200 is false. Shade the side away from (0, 0), above the line.
Step 4. Check (20, 12): 5(20) + 10(12) = 100 + 120 = 220, and 220 \ge 200.
Answer: 5s + 10a \ge 200. Yes, 20 student tickets and 12 adult tickets raise $220, which is enough. Only whole-number points in the shaded region make sense here, because you can’t sell part of a ticket.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Wrong type of line | A solid line for y \lt x + 2 | Use a dashed line for < and >, and a solid line for ≤ and ≥. |
| Not flipping the sign | -3y \gt -2x + 6 becomes y \gt \frac{2}{3}x - 2 | Dividing by −3 flips the sign: y \lt \frac{2}{3}x - 2. |
| Using the shortcut too early | Shading above for -y \gt x | The shortcut only works once y is alone with a positive coefficient. Test a point. |
| Testing a point on the line | Testing (0, 0) for y \lt 2x | (0, 0) is on y = 2x. Pick another point, like (0, 1). |
Graphing More Than One Inequality
When you graph two or more inequalities on the same grid, the solutions are the points in the overlap of the shaded regions. You’ll learn this in the next lesson, Systems of Inequalities.
Practice Problems
For problems 1 and 2, decide if the point is a solution. For problems 3–9, describe the graph: the boundary line, solid or dashed, and which side to shade.
- (1, 4); y \gt 2x + 1
- (−2, 0); y \le -x - 3
- y \lt x + 2
- y \ge -2x + 1
- y \gt \frac{3}{4}x - 3
- 3x + y \le 4
- x - 2y \lt 6
- x \ge -3
- y \lt 4
- Pens cost $2 and notebooks cost $4. You can spend at most $20. Write an inequality. Can you buy 4 pens and 3 notebooks?
Show answers
- Yes. 4 \gt 2(1) + 1 means 4 \gt 3, which is true.
- No. 0 \le -(-2) - 3 means 0 \le -1, which is false.
- Dashed line y = x + 2. Test (0, 0): 0 \lt 2 is true. Shade below the line, the side with (0, 0).
- Solid line y = -2x + 1. Test (0, 0): 0 \ge 1 is false. Shade above the line.
- Dashed line y = \frac{3}{4}x - 3. Test (0, 0): 0 \gt -3 is true. Shade above the line, the side with (0, 0).
- Rewrite as y \le -3x + 4. Solid line y = -3x + 4. Test (0, 0): 0 \le 4 is true. Shade below the line.
- -2y \lt -x + 6, so y \gt \frac{1}{2}x - 3 (flip the sign). Dashed line y = \frac{1}{2}x - 3. Test (0, 0): 0 \lt 6 is true. Shade above the line.
- Solid vertical line x = -3. Shade to the right.
- Dashed horizontal line y = 4. Shade below.
- 2p + 4n \le 20. Check (4, 3): 2(4) + 4(3) = 20, and 20 \le 20 is true. Yes, you spend exactly $20, so the point is on the solid boundary line.
Graphing Linear Inequalities FAQ
How do you know whether to shade above or below the line?
Test a point that isn’t on the line, such as (0, 0). If it makes the inequality true, shade its side. If it makes it false, shade the other side.
When do you use a dashed line?
Use a dashed line for < and >, because points on the line are not solutions. Use a solid line for ≤ and ≥.
Why do you flip the inequality sign?
Multiplying or dividing both sides by a negative number reverses the order of numbers. For example, 2 \lt 5, but -2 \gt -5. So the sign must flip to stay true.
What is the solution of a linear inequality in two variables?
The solution is every point in the shaded region. If the line is solid, the points on the line are solutions too.
Summary
- Graph the boundary line first, using = in place of the inequality sign.
- Dashed line for < or >. Solid line for ≤ or ≥.
- Test a point not on the line, then shade the side that makes the inequality true.
- Flip the sign when you multiply or divide by a negative number.
- When y is alone, > and ≥ shade above, and < and ≤ shade below.
Previous lesson: Systems of Equations Word Problems
Next lesson: Systems of Inequalities
Back to the Algebra 1 Index