Graph y <= 2x + 1 and y > -x

Step by step

The question

Question: graph y ≤ 2x + 1 and y > −x − 2.

① y ≤ 2x + 1 and ② y > −x − 2

We shade the answer of each one.

Where both are shaded, both are true.

answer = the part shaded twice

1 Graph ①

Draw the grid.

①: The sign has "or equal to".

So the line is solid.

① ≤ → solid line

b = 1.

Start on the y-axis at 1.

m = 2.

Go right 1,

then up 2.

Dot.

Join the dots: solid.

Which side?

We test one point: (0, 0).

Put x = 0 and y = 0 into ①.

① 0 ≤ 2(0) + 1

0 ≤ 1 true

Is 0 less than or equal to 1?

Yes, true.

So we shade the side with (0, 0), in red.

2 Graph ②

②: The sign is more than, with no "or equal to".

So the line is dashed.

② > → dashed line

b = −2.

Start on the y-axis at −2.

m = −1.

Go right 1,

then down 1.

Dot.

Join the dots: dashed.

Which side?

We test one point: (0, 0).

Put x = 0 and y = 0 into ②.

② 0 > −(0) − 2

0 > −2 true

Is 0 more than −2?

Yes, true.

So we shade the side with (0, 0), in blue.

3 Where both are shaded

This part is shaded twice.

It is the answer.

Careful!

The answer is not everything we shaded.

Only the part shaded twice works in both.

Careful: not every shaded part ✗

4 Check

Let's check with a point in the purple part: (1, −1).

Put it into ①.

① −1 ≤ 2(1) + 1

−1 ≤ 3 true

Now ②.

② −1 > −1 − 2

−1 > −3 true

True in both!

So the answer is right.

Let's look back.

First: graph ①.

Then: graph ②.

Then: where both are shaded.

Last: check.

Answer