Grade by Grade
C. Algebra · Linear and Non-Linear Relations

Inequalities: Graphing Regions

C4.2 A1About 70 minutes
Learning goals
  • I can graph the region for inequalities such as y > k, x ≤ k, x + y ≤ k, x − y > k, ax + by ≥ k and xy ≥ k.
  • I can decide whether the boundary is solid (included) or dashed (not included).
  • I can use a test point to decide which side of the boundary to shade.
  • I can use an inequality to model a budget and check whether a combination is possible.

Words to know

inequality
A statement that uses <, >, ≤ or ≥ to compare two expressions.
region
The part of the coordinate plane that contains all the points that make an inequality true. It is shown by shading.
boundary
The line or curve where the inequality becomes an equation. It separates the points that work from the points that do not.
test point
A point not on the boundary, such as (0, 0), that you substitute to find out which side of the boundary to shade.

Remember from Grade 8

In Grade 8 you solved inequalities with one variable, like 2x − 3 < 7, and graphed the answer on a number line: an open circle for < or > and a closed circle for ≤ or ≥. Now the inequalities have two variables, so the answers are not just points on a line: they fill a whole region of the coordinate plane.

Party snacks on a budget

You have at most $24 for a party. Chips cost $3 a bag and drinks cost $2 a bottle. If c is the number of bags of chips and d the number of drinks, then

3c + 2d ≤ 24

There are lots of combinations that work: 4 bags and 5 drinks costs $22, which is fine. 6 bags and 4 drinks costs $26, which is too much. All the combinations that work form a region on a graph.

The three steps

  1. Draw the boundary. Replace the inequality sign with = and graph that line or curve.
  • Use a solid line for ≤ or ≥: points on the boundary are included.
  • Use a dashed line for < or >: points on the boundary are not included.
  1. Pick a test point that is not on the boundary. (0, 0) is easiest when the boundary does not pass through it.
  2. Shade the side that contains the test point if it makes the inequality true. Otherwise, shade the other side.

For y > k or y < k you can skip the test: > means above the horizontal line and < means below. For x > k shade to the right of the vertical line, and for x < k to the left.

−4−224−4−2240xyy > 2y > 2: dashed line, shade above−4−224−4−2240xyx ≤ −1x ≤ −1: solid line, shade left
Left: y > 2 is everything above the dashed line y = 2. The open circle at (0, 2) is a reminder that points on the dashed line, like (0, 2), are not included. Right: x ≤ −1 is everything to the left of the solid line x = −1, including the line itself.
Worked example

Example 1: Graph x + y ≤ 4

Graph the region x + y ≤ 4.

  1. Boundary: x + y = 4, with intercepts (4, 0) and (0, 4). The sign is ≤, so draw a solid line.
  2. Test (0, 0): 0 + 0 = 0, and 0 ≤ 4 is true.
  3. Shade the side of the line that contains (0, 0): below and to the left.
  4. Extra check with a point on the other side, (4, 3): 4 + 3 = 7, and 7 ≤ 4 is false ✓.

A solid line through (4, 0) and (0, 4), with the region below and to the left shaded.

−2−1123456−2−11234560xy(0, 0): 0 ≤ 4 ✓x + y = 4(4, 3): 7 ≤ 4 ✗x + y ≤ 4
x + y ≤ 4: the solid boundary x + y = 4 is included. The test point (0, 0) works, so its side is shaded; (4, 3) does not work and lies on the unshaded side.
Worked example

Example 2: Graph 2x − 3y > 6

Graph the region 2x − 3y > 6.

  1. Boundary: 2x − 3y = 6. y = 0 gives x = 3; x = 0 gives −3y = 6, so y = −2. Intercepts (3, 0) and (0, −2). The sign is >, so draw a dashed line.
  2. Test (0, 0): 2(0) − 3(0) = 0, and 0 > 6 is false.
  3. So shade the side without the origin: below and to the right of the line.
  4. Check (5, 0): 10 − 0 = 10 > 6 ✓. That point is below and to the right, so the shading is correct.

A dashed line through (3, 0) and (0, −2), with the region below and to the right shaded.

Worked example

Example 3: A region bounded by a curve

The sides x and y (in metres) of a rectangular patio must give an area of at least 4 m²: xy ≥ 4 with x > 0 and y > 0. Graph the region.

  1. Boundary: the hyperbola xy = 4 in the first quadrant, through (1, 4), (2, 2) and (4, 1). The sign is ≥, so it is solid.
  2. Test (1, 1): 1 × 1 = 1, and 1 ≥ 4 is false. So (1, 1), close to the origin, is not in the region.
  3. Test (4, 4): 16 ≥ 4 is true. Shade the side of the curve away from the origin.

The region above and to the right of the curve xy = 4, including the curve.

01234567812345678xy(1, 1): 1 ≥ 4 ✗(4, 4): 16 ≥ 4 ✓xy = 4xy ≥ 4
xy ≥ 4 in the first quadrant: the shaded region is on the far side of the curve from the origin. Every point there is a patio with an area of 4 m² or more.
Worked example

Example 4: Back to the party

Use 3c + 2d ≤ 24. (a) Can you buy 4 bags of chips and 5 drinks? (b) 6 bags and 4 drinks? (c) What is the largest number of drinks you can buy if you buy 2 bags of chips?

  1. (a) 3(4) + 2(5) = 12 + 10 = 22, and 22 ≤ 24 ✓. Yes.
  2. (b) 3(6) + 2(4) = 18 + 8 = 26, and 26 ≤ 24 is false. No.
  3. (c) 3(2) + 2d ≤ 24, so 2d ≤ 18 and d ≤ 9. 9 drinks.
  4. In this context c ≥ 0 and d ≥ 0, and only whole numbers make sense, so the possible combinations are the whole-number points in the triangle with corners (0, 0), (8, 0) and (0, 12).

(a) Yes, $22. (b) No, $26. (c) 9 drinks.

Shortcut for y > or y <

If an inequality is solved for y, like y < 2x + 1, you do not need a test point: y > means shade above the line and y < means shade below it. This only works when y is alone on the left side.

Just for fun

Why was the dashed line so relaxed at the party?

It knew it wasn't included in anything.

Hands-on activity

Flyer budget map

You need: a grocery or store flyer (paper or online), graph paper, a pencil, a coloured pencil.

  1. Choose two items from the flyer and a budget (for example, $30).
  2. Write an inequality: (price of item 1)x + (price of item 2)y ≤ budget.
  3. Find the intercepts and draw the solid boundary line in the first quadrant. Shade the region.
  4. List three combinations inside the region and one just outside. Check each one with the inequality.
  5. Which combination spends the most money without going over? Is it on the line or just inside?

What to look for: points inside or on the line are affordable. The best combinations are usually whole-number points close to the boundary line.

Your turn

1

Describe the region y < −1.

2

Describe the region x ≥ 3.

3

Is (2, 5) in the region x + y ≤ 6?

4

Is (−1, −4) in the region x − y > 2?

5

Should the boundary of 3x + 2y < 12 be solid or dashed?

6

Does the region x + y ≥ 5 contain the origin?

7

Is (2, 3) in the region xy > 4? Is (1, 3)?

More practice: warm-up

1

Which symbol means "at most"?

2

Describe the region y ≥ 0.

3

What does a dashed boundary mean?

4

Is (0, 0) in the region y > 2?

5

Is (5, 1) in the region x > 4?

More practice: on your own

1

Describe how to graph x − y < 3.

2

Describe how to graph 4x + 5y ≥ 20.

3

Write an inequality for the region below the solid line y = 2x + 1.

4

A $60 gift card buys songs at $1.50 and albums at $12: 1.5s + 12a ≤ 60. Can you buy 2 albums and 20 songs? 3 albums and 18 songs?

5

Describe the region xy < 0.

Challenge

1

Describe the region where x ≥ 0, y ≥ 0 and x + y ≤ 5 are all true. How many points with whole-number coordinates are in it?

2

For positive x and y, which side of the curve xy = 6 is the region xy ≤ 6? Is (2, 4) in it?

3

For the party you need at least 4 drinks, and you still spend at most $24 (chips $3, drinks $2). Write the inequalities and find the greatest number of bags of chips you can buy.

Check your understanding

Quiz

1.

Which inequality is shown by a dashed horizontal line through (0, 3) with the region below shaded?

2.

Which point is in the region x + y > 6?

3.

To graph 2x + y ≤ 8, you test (0, 0). What do you shade?

4.

The boundary for x − y ≥ 1 should be:

5.

Which describes the region xy > 0?

6.

Chips $3 and drinks $2, at most $24. Which combination is not possible?

For parents and teachers

Big idea: Expectation C4.2 includes graphing the inequalities associated with x = k, y = k, x + y = k, x − y = k, ax + by = k and xy = k, and identifying the regions they define. The three steps are: draw the boundary (solid or dashed), test a point, shade.

Common difficulties:

  • Solid versus dashed. Link it to the number-line circles from Grade 8: closed circle = solid line (≤, ≥), open circle = dashed line (<, >).
  • Using the "above/below" shortcut when y is not alone, or when the inequality has a negative coefficient on y. The test point always works.
  • Testing a point that is on the boundary. Choose another point.

How to help: Budgets are a natural context: ask your child to write an inequality for two things you buy often and check a few combinations together.