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

Step by step

The question

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

We shade the answer of each one.

Where both are shaded, both are true.

1 Crossing points of ①

On the x-axis, y = 0.

Put 0 in place of y.

Adding 0 changes nothing.

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

2 divided by 2 is 1.

So x = 1.

On the y-axis, x = 0.

Put 0 in place of x.

2 times 0 is 0.

0 plus y is just y.

So y = 2.

So line ① crosses the axes at (1, 0) and (0, 2).

2 Crossing points of ②

On the x-axis, y = 0.

Put 0 in place of y.

1 times 0 is 0.

Taking away 0 changes nothing.

So x = 1.

On the y-axis, x = 0.

Put 0 in place of x.

0 minus y is −y.

−y means −1 times y.

We want just y.

To remove times −1, we divide by −1.

We do it on both sides, so they stay equal.

−1 divided by −1 is 1.

So just y is left.

Plus over minus is minus.

1 divided by −1 is −1.

So y = −1.

So line ② crosses the axes at (1, 0) and (0, −1).

3 Graph ①

Draw the grid.

①: The sign is more than, with no "or equal to".

So the line is dashed.

Put the two crossing points on the grid,

and join them: dashed.

Which side?

We test one point: (0, 0).

Put x = 0 and y = 0 into ①.

Is 0 more than 2?

No, false.

So we shade the other side, away from (0, 0), in red.

4 Graph ②

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

So the line is dashed.

Put the two crossing points on the grid,

and join them: dashed.

Which side?

We test one point: (0, 0).

Put x = 0 and y = 0 into ②.

Is 0 less than 1?

Yes, true.

Careful!

The sign alone does not say above or below.

When y has a minus in front, it can turn.

The test point tells us.

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

5 Where both are shaded

This part is shaded twice.

It is the answer.

6 Check

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

Put it into ①.

Now ②.

True in both!

So the answer is right.

Let's look back.

First: crossing points of ①.

Then: crossing points of ②.

Then: graph ①.

Then: graph ②.

Then: where both are shaded.

Last: check.