Inequalities and regions

The Maths Habit Handout to print

Draw these lines.

(a) x = −4

xy−5−4−3−2−112−3−2−1123a vertical line through (−4, 0)

(b) y = −2

xy−3−2−1123−3−11230a horizontal line through (0, −2)

(c) y = 1 − 3x

xy−2−1123−3−2−12340through (0, 1) and (1, −2)

Give three points whose coordinates add up to less than 3.

(0, 0), (5, −4), (−1, 3.5) … All of them are on one side of a line.

xy−5−4−3−2−112345−5−4−3−2−112450

Region
An area of the grid.
Boundary line
The edge of a region.
Dashed line
For < or >. Points on it are not included.
Solid line
For ≤ or ≥. Points on it are included.

Show why

x < 2
−101234xy−11234−2−1120
x ≤ 2
−101234xy−11234−2−1120

Open circle, dashed line: not included. Closed circle, solid line: included. The same rule.

1Dashed or solid?2Draw the line.3Test a point not on the line.4True: that side. False: the other side.5Shade only the region where every inequality is true. Label it R.

Show x + 2y < 8 on the grid.

xy−2−112345678−2−112350
1Dashed: <.
2Through (8, 0) and (0, 4).
3Test (0, 0): 0 + 0 < 8. True.
4Shade the side with (0, 0).

Show y ≥ 2x − 3 on the grid.

xy−3−2−1123−4−2−11230
1Solid: ≥.
2Through (0, −3) and (2, 1).
3Test (0, 0): 0 ≥ −3. True:
4the side with (0, 0).

Shading the other side and leaving R clear also scores. We shade the region we want.

1Dashed or solid?2Draw the line.3Test a point not on the line.4True: that side. False: the other side.5Shade only the region where every inequality is true. Label it R.

Show x + 2y < 8 on the grid.

xy−2−112345678−2−112350
1Dashed: <.
2Through (8, 0) and (0, 4).
3Test (0, 0): 0 + 0 < 8. True.
4Shade the side with (0, 0).

Show x > −3 on the grid.

xy−5−4−3−2−1123−3−2−1123
1Dashed.
2A vertical line through (−3, 0).
3Test (0, 0): 0 > −3. True:
4the right of the line.
1Dashed or solid?2Draw the line.3Test a point not on the line.4True: that side. False: the other side.5Shade only the region where every inequality is true. Label it R.Points on a dashed line are not in R. Points on a solid line are.

Shade the region R where x > 1, y ≥ 2 and x + 2y < 10. List the points with integer coordinates in R.

Rxy−112345678910−113460
1x = 1 dashed, y = 2 solid, x + 2y = 10 dashed through (10, 0) and (0, 5).
5R has corners (1, 2), (6, 2) and (1, 4.5).
(2, 2), (2, 3), (3, 2), (3, 3), (4, 2), (5, 2)
(4, 3) is on the dashed line x + 2y = 10, so it is not in R.

Shade the region R where y ≥ 0, x < 3 and y < x. List the points with integer coordinates in R.

xy−11234−11234
3(0, 0) is on y = x, so test (2, 0): 0 < 2, true.
(1, 0), (2, 0), (2, 1)
(1, 0) and (2, 0) are on the solid line y = 0, so they are in.
1Dashed or solid?2Draw the line.3Test a point not on the line.4True: that side. False: the other side.5Shade only the region where every inequality is true. Label it R.Points on a dashed line are not in R. Points on a solid line are.

Shade the region R where x > 1, y ≥ 2 and x + 2y < 10. List the points with integer coordinates in R.

Rxy−112345678910−113460
1x = 1 dashed, y = 2 solid, x + 2y = 10 dashed through (10, 0) and (0, 5).
5R has corners (1, 2), (6, 2) and (1, 4.5).
(2, 2), (2, 3), (3, 2), (3, 3), (4, 2), (5, 2)
(4, 3) is on the dashed line x + 2y = 10, so it is not in R.

Shade the region R where y ≥ −x, y < 2 and x < 2. List the points with integer coordinates in R.

xy−3−2−1123−3−2−130
5R has corners (−2, 2), (2, 2) and (2, −2)
(−1, 1), (0, 0), (0, 1), (1, −1), (1, 0), (1, 1)
(0, 0) is on the solid line y = −x, so it is in.
Find each line's equation. Test a point inside R in each one.

Write down the three inequalities that describe R.

Rxy−2−112345−3−1134
x = −1, y = −2, x + y = 2.
Test (0, 0): 0 > −1, 0 ≥ −2, 0 ≤ 2.
x > −1, y ≥ −2, x + y ≤ 2.

Write down the three inequalities that describe R.

Rxy−3−2−112345−212456
x = −2: x ≥ −2.
y > −1, 3x + 2y ≤ 6.
The line x = 4 is not an edge of R.
Find each line's equation. Test a point inside R in each one.

Write down the three inequalities that describe R.

Rxy−2−112345−3−1134
x = −1, y = −2, x + y = 2.
Test (0, 0): 0 > −1, 0 ≥ −2, 0 ≤ 2.
x > −1, y ≥ −2, x + y ≤ 2.

Early finishers. Write down the three inequalities that describe R.

Rxy−3−2−1123−5−4−3−2−11240
y > −2x, y ≤ 3, x < 2.
(0, 0) is on y = −2x. Test (1, 0).

The line y = 3x − 4 is drawn dashed. Region R is on the side of the line that contains the point (0, 2). Which inequality describes R?

xy−1123−3−2−1130
Ay > 3x − 4
By < 3x − 4
Cy ≥ 3x − 4
Dy = 3x − 4
A: rightB: test the point: 2 < −4 is falseC: dashed means not included: > not ≥D: R is a region, so the answer is an inequality

Inequalities and regions worksheet, page 1

Inequalities and regions worksheet, page 2

Answer sheet

Hide a point

In pairs. A picks a point with integer coordinates and writes it down, hidden. B asks for inequalities ("Is x > 0 true?"). A answers true or false, and B shades on a grid what is left, until one point remains.

Clues: x > 0, y < 0, x + y ≥ 1, x ≤ 2, y ≥ −1. Which point is it? (2, −1)

  1. Shade the region R where y > −1, x ≤ 3 and y ≤ x + 3.

    corners (−4, −1), (3, −1), (3, 6)
    −424−2246

  2. Write down the three inequalities that describe R.

    y ≥ 1, x > −1, 2x + y ≤ 5
    R−226

  3. List the points with integer coordinates in the region where y > 0, x ≥ −1 and x + y ≤ 2.

    (−1, 1), (−1, 2), (−1, 3), (0, 1), (0, 2), (1, 1)
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