Lines and regions

A delivery zone, a speed limit, a budget: often the answer is not one point but a whole region of points that work.

You can graph a line from its rule, such as

y = 2x + 1

. Now: what about

y < 2x + 1

? Every point whose

y

is less than

2x + 1

makes it true, and those points fill a whole region of the graph.

You also meet lines with only one letter:

x = 3

(vertical) and

y = -2

(horizontal).

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

In or out?

Put $x = 1$ and $y = 4$ into $y < 2x + 1$ . Is it true?

Is the point

(1, 4)

in the region

y < 2x + 1

?Yes: 4 is bigger than 1No: 4 is not less than 2 × 1 + 1 = 3It is on the boundary line

TryTry it on the board and compare the result with your prediction.

Into the region

Try a point on the dashed line too. Does it make $y < 2x + 1$ true?

Try it yourselfYour edits are saved in this browser.

Drag $P$ into the shaded region $y < 2x + 1$ . Check its coordinates in the inequality. Was your prediction right about $(1, 4)$ ?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Boundary, then test

Here is how to graph an inequality.

Worked exampleBefore each step, say what you think comes next.

$y < 2x + 1$ The boundary is the line $y = 2x + 1$ .
$\text{dashed}$ $<$ does not include the line's points (for $\le$ or $\ge$ , draw it solid).
$(0, 0):\; 0 < 2 \times 0 + 1$ Test a point not on the line. $0 < 1$ is true.
$\text{shade the side with } (0, 0)$ True: shade the test point's side. (False: shade the other side.)
$x = 3,\; y = -2$ One letter: $x = 3$ is vertical (every point has $x$ = 3); $y = -2$ is horizontal.

Solid or dashed for $y \ge x - 2$ ?

Read Ana's working. Which line does not match the sign?

Need a hint?

The orange region is x ≥ 3: on or right of x = 3.

Another hint

The other is y < 2: below the dashed line y = 2.

Try it yourselfYour edits are saved in this browser.

For

y \ge x - 2

, is the boundary line solid or dashed?Solid: ≥ includes the points on the lineDashed: points on the line are not includedThere is no boundary

Ana graphs $y > x + 1$ . Click the line where her working goes wrong.

Two regions: $x \ge 3$ and $y < 2$ . Drag $P$ to a point that is in both.

Use the board. This ticks when you get there.

ModifyChange the problem on purpose and see what happens.

One change at a time

PractiseA few on your own, to check you have it.

Regions

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Fit the region

Find two different answers. What do they have in common?

Try it yourselfYour edits are saved in this browser.

Need a hint?

(0, 0) in y ≤ mx + c means 0 ≤ c.

Another hint

(0, 3) outside means c is less than 3. Then check B (2, 1): 1 ≤ 2m + c.

Set $m$ and $c$ so the region $y \le mx + c$ contains $A(0, 0)$ and $B(2, 1)$ but not $C(0, 3)$ .

Use the board. This ticks when you get there.

Lines and regions

In or out?

Into the region

Boundary, then test

One change at a time

Regions

Fit the region

Students will be able to

Before class

Running the lesson

When students get stuck

Stretch and support

Where it leads

What each step is for