Worked examples
The question
Question: graph y > 2x − 1 and y ≤ −x + 3.
① y > 2x − 1 and ② y ≤ −x + 3
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 is more than, with no "or equal to".
So the line is dashed.
① > → dashed line
b = −1.
Start on the y-axis at −1.
m = 2.
Go right 1,
then up 2.
Dot.
Join the dots: dashed.
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 more than −1?
Yes, true.
So we shade the side with (0, 0), in red.
2 Graph ②
②: The sign has "or equal to".
So the line is solid.
② ≤ → solid line
b = 3.
Start on the y-axis at 3.
m = −1.
Go right 1,
then down 1.
Dot.
Join the dots: solid.
Which side?
We test one point: (0, 0).
Put x = 0 and y = 0 into ②.
② 0 ≤ −(0) + 3
0 ≤ 3 true
Is 0 less than or equal to 3?
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) + 3
−1 ≤ 4 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.
▶ Watch it solved
The question
Question: graph y < 3 and y > x − 1.
① y < 3 and ② y > x − 1
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 is less than, with no "or equal to".
So the line is dashed.
① < → dashed line
b = 3.
Start on the y-axis at 3.
A flat line through the dot: dashed.
Which side?
We test one point: (0, 0).
Put x = 0 and y = 0 into ①.
① 0 < 3 true
Is 0 less than 3?
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 = −1.
Start on the y-axis at −1.
m = 1.
Go right 1,
then up 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 − 1
0 > −1 true
Is 0 more than −1?
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 < 3 true
Now ②.
② −1 > (−1) − 1
−1 > −2 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.
▶ Watch it solved
The question
Question: graph x − y > 3 and y < 2.
① x − y > 3 and ② y < 2
We shade the answer of each one.
Where both are shaded, both are true.
answer = the part shaded twice
1 Crossing points of ①
On the x-axis, y = 0.
Put 0 in place of y.
x − 1 × 0 = 3
1 times 0 is 0.
Taking away 0 changes nothing.
So x = 3.
x = 3
On the y-axis, x = 0.
Put 0 in place of x.
0 − y = 3
0 minus y is −y.
−y = 3
−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.
−y−1 = 3−1
−1 divided by −1 is 1.
So just y is left.
Plus over minus is minus.
3 divided by −1 is −3.
3−1 = −3
So y = −3.
So line ① crosses the axes at (3, 0) and (0, −3).
y = −3
2 Graph ①
Draw the grid.
①: The sign is more than, with no "or equal to".
So the line is dashed.
① > → dashed line
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.
① 0 − 0 > 3
0 > 3 false
Is 0 more than 3?
No, false.
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.
Careful: > does not always mean above
So we shade the other side, away from (0, 0), in red.
3 Graph ②
②: The sign is less than, with no "or equal to".
So the line is dashed.
② < → dashed line
b = 2.
Start on the y-axis at 2.
A flat line through the dot: dashed.
Which side?
We test one point: (0, 0).
Put x = 0 and y = 0 into ②.
② 0 < 2 true
Is 0 less than 2?
Yes, true.
So we shade the side with (0, 0), in blue.
4 Where both are shaded
This part is shaded twice.
It is the answer.
5 Check
Let's check with a point in the purple part: (1, −3).
Put it into ①.
① 1 − (−3) > 3
4 > 3 true
Now ②.
② −3 < 2 true
True in both!
So the answer is right.
Let's look back.
First: crossing points of ①.
Then: graph ①.
Then: graph ②.
Then: where both are shaded.
Last: check.
▶ Watch it solved
