Before you read

Answer first — see what you already know.

Which point is in the solution set of $y > x + 2$ ?

A
$(0, 1)$
B
$(0, 2)$
C
$(0, 3)$
D
$(0, - 1)$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The graph of $y \leq - x + 5$ has a boundary line. Is the boundary line dashed or solid?

A
Dashed, because $y$ can equal $- x + 5$
B
Solid, because $y$ can equal $- x + 5$
C
Dashed, because the inequality is strict
D
It depends on the region

Check your answer

Answer:

Theory

Graphing an Inequality in Two Variables

To graph $y < 2x + 3$ (or $\leq$ , $>$ , $\geq$ ):

Step 1: Draw the boundary line — graph $y = 2x + 3$ .
- Dashed line for $<$ or $>$ (boundary NOT included)
- Solid line for $\leq$ or $\geq$ (boundary included)

Boundary line y = x - 2 for the inequality y > x - 2

Step 2: Shade the correct side.
- For $y <$ or $y \leq$ : shade below the line
- For $y >$ or $y \geq$ : shade above the line

Shortcut: if the inequality is solved for $y$ :
- $y < ...$ $\to$ shade below
- $y > ...$ $\to$ shade above

Theory

The Test Point Method

If you're unsure which side to shade:
1. Pick a test point not on the line ( $(0, 0)$ is easiest if the line doesn't pass through the origin)
2. Substitute into the inequality
3. If the result is true: shade the side containing that point
4. If false: shade the other side

Example 1

Determine which side of $2x + y > 6$ contains the solutions.

Test $(0, 0)$ : $2(0) + 0 = 0 > 6$ ? $\to$ $0 > 6$ is false.

So $(0, 0)$ is NOT in the solution set.

Shade the side opposite to $(0, 0)$ — the side above/right of the line.

Shade the side away from the origin.

Continue this lesson in the app

3 more sections including examples, practice problems, and step-by-step solutions.

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