Part 20 of 20 · InequalitiesSee the whole course
How to graph an inequality (dashed or solid, and which side to shade)
Graph the boundary line first: dashed for > and <, solid for ≥ and ≤. Then pick a test point and shade the side where it makes the inequality true.
On a number line
For one letter, the graph is a ray. x > 2 has an open circle at 2 and an arrow pointing right: 2 itself is not included. x ≥ 2 has a closed circle, because 2 is included. For x < 2 and x ≤ 2 the arrow points left.
On a coordinate plane: three steps
- Draw the boundary. Pretend the sign is an equals sign and graph the line. Make it dashed for > or < (the line is not part of the answer) and solid for ≥ or ≤ (it is).
- Pick a test point that is not on the line. (0, 0) is the easiest, if the line does not pass through it.
- Put it in. If the statement is true, shade the side with that point. If it is false, shade the other side.
Example 1: y > x + 1
Boundary: y = x + 1, drawn dashed because the sign is >.
Test (1, 4): is 4 > 1 + 1? 4 > 2 is true, so shade the side with (1, 4): above the line.
Test (3, 0): is 0 > 3 + 1? 0 > 4 is false, so that side stays blank.
Try it below. Move the test point and watch it turn green inside the zone and red outside. Then switch between >, ≥, < and ≤.
- boundary: dashed line y = x + 1
- test (1, 4): 4 > x + 1 → 4 > 2: true
Example 2: y < 2x − 1
Boundary: y = 2x − 1, dashed.
Test (0, 0): is 0 < 2(0) − 1? 0 < −1 is false, so shade the other side: below the line.
Check with (0, −3): −3 < −1 is true ✓.
Example 3: y ≤ −x + 3
Boundary: y = −x + 3, solid because ≤ includes the line.
Test (0, 0): is 0 ≤ 3? True, so shade the side with the origin: below the line.
Why use a test point?
You may have heard “shade above for >, below for <”. That works only when y is on its own and positive. A test point works for every inequality, even one written like −y > x − 1, where dividing by −1 flips the sign (see how to solve a linear inequality).
Watch it explained in under a minute
The four mistakes to avoid
- Drawing a solid line for > or <. A strict inequality leaves the line out, so it is dashed.
- Shading the wrong side. Do not guess: test a point.
- Skipping the test point and trusting “above means greater”.
- Testing a point that lies on the line. It gives equal, which tells you nothing about either side.
Try it yourself
Tap a question to see the answer.
1. Graph y > 2x
Dashed line y = 2x. (0, 0) is on the line, so test (0, 3) instead: 3 > 0 is true. Shade above the line.
2. Graph y ≤ −x + 4
Solid line y = −x + 4. Test (0, 0): 0 ≤ 4 is true. Shade the side with the origin, below the line.
3. Graph y < x − 3
Dashed line y = x − 3. Test (0, 0): 0 < −3 is false, so shade the other side, below the line. Check (0, −5): −5 < −3 is true ✓.
Where this fits in
That completes the first course: functions, lines, equations and inequalities, in order. You can browse all the posts in the series index, or revisit slope-intercept form, the idea that most of these posts are built on.