Start with a point. Build a line. Find a whole region.

From points to shaded regions

Find a point
Build a line
Shade a region
Try it yourself

Your coordinate map1 square = 1 unit

x: acrossy: up or downsolution region

The two number lines cross at (0, 0), the origin.

1 of 10 · Find a point

Every point has an address.

The horizontal number line is the x-axis. The vertical one is the y-axis. Start where they cross: the origin (0, 0).

(2, 3)

Read x first, then y. Count the spaces between grid lines.

First, move 2 spaces right from the origin.

Now find a negative address.

Negative x means left. Negative y means down. Plot (−2, −1).

Start at (0, 0). Go 2 left, then 1 down.

Move the crosshair with the arrow buttons, or tap a grid crossing.

A rule gives us points.

y = 2x + 1

This says: choose an x, multiply it by 2, then add 1 to find y.

Choose x → multiply by 2 → add 1

x	Find y	Point (x, y)
−1	2 × (−1) + 1 = −1	(−1, −1)
0	2 × 0 + 1 = 1	(0, 1)
1	2 × 1 + 1 = 3	(1, 3)

Watch each table row turn into a point on the grid.

Put a ruler through the points.

These points line up. Draw one straight line through them and extend it in both directions.

The repeated move is right 1, then up 2. This line’s slope is 2.

The points mark places on the line. They are not its ends.

Does the boundary count?

y > 2x + 1

> means greater than. The line y = 2x + 1 is the boundary: it separates the two sides.

At (1, 3), the two sides are equal.
3 > 2 × 1 + 1
3 > 3 is false.

Points on the boundary do not count. Which line shows that?

Greater y means higher up.

Keep x = 1. The boundary is at y = 3. Compare a point above it and a point below it.

(1, 4): 4 > 3 is true — above.
(1, 2): 2 > 3 is false — below.

One symbol changes the region.

Try all four symbols. The boundary stays in the same place.

>: shade above; use a dashed line.

The line is excluded because equality does not count.

With y alone on the left:
greater → above; less → below.
“Or equal to” → a solid boundary.

Explore all 4 symbols to continue.

Build your own shaded region.

y ≤ −x + 2

First use y = −x + 2 to find the boundary. Here −x means the opposite of x.

Find a point: if x = 1, then y = −1 + 2.

y =

Check each part of your graph.

y ≤ −x + 2

Find the opposite of x, then add 2.

x = 0: y = −0 + 2 = 2 → (0, 2)
x = 1: y = −1 + 2 = 1 → (1, 1)
x = 2: y = −2 + 2 = 0 → (2, 0)

Solid: ≤ includes equality.
The boundary points are part of the solution.

Below: at x = 0, the boundary is y = 2. A lower point, (0, 0), works:
0 ≤ −0 + 2 → 0 ≤ 2 is true.

You can graph an inequality.

A line records equality. A shaded region shows all the points that make the inequality true.

Make a table. Choose x and find y.
Plot. Start at the origin; x first, then y.
Draw the boundary. Straight through the points, extending both ways.
Style and shade. Equality included? Solid. Greater y? Above. Less y? Below.

Use the above/below rule when y is alone on the left. Check one point if you are unsure.