Year 10 Extended · Unit 2 Generalizing, coordinate geometry and linear programming · Lesson 5 of 6

Which points of the plane are allowed?

An inequality splits numbers, or points of the plane, into those that make it true and those that do not. This lesson finds the reasons behind the rules for solving one, and the region where several hold at once.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8: solve 2x + 7 < 15.
    Answerx < 4. Subtract 7, then divide by 2.
  2. From Year 9: what is the gradient of the line y = 5 − 2x?
    Answer−2. The coefficient of x.
  3. From lesson 4: what does elimination leave when two equations describe the same line?
    Answer0 = 0, and every point on the line is a solution.
Words used in this lesson
inequality
a statement that one expression is larger or smaller than another. For example: x + 4 < 10.
strict inequality
one that uses < or > so the end value is not included. For example: x > 0.
boundary line
the line where the two sides of an inequality meet. For example: y = 3x in y ≤ 3x.
region
the set of points that satisfy every inequality in a system. For example: the triangle inside three lines.
test point
a point not on the boundary that you try in the inequality. For example: (0, 0) when it is not on the line.
double inequality
two inequalities written together. For example: 3 < x ≤ 8.

Allowed and not allowed

A community centre in Cape Town has a hall that holds at most 60 people, and the organizers want at least twice as many students as adults. Each of these rules cuts the possible (adults, students) pairs into two sets: allowed and not allowed. Write each rule as an inequality in a (adults) and s (students): together they leave a region.

Predict first: for the inequality 6x > 18, the solutions are all the numbers on one side of a boundary value. Which side? Now change the 6 to a −6 in your head: does the side change? Write a prediction, then use the first figure.

2
0
x > 6the solutions
falseis the test value a solution

2x + 3 > 15: subtract 3 to get 2x > 12, then divide by 2: x > 6. At x = 0: 2 × 0 + 3 = 3, which is not greater than 15, so false.

Show me the steps
  1. 4x + 3 > 15: subtract 3 to get 4x > 12, divide by 4: x > 3. At x = 5: 4 × 5 + 3 = 23, greater than 15, so true.
  2. −2x + 3 > 15: subtract 3 to get −2x > 12. Dividing by −2 reverses the sign: x < −6. At x = −8: −2 × (−8) + 3 = 19, greater than 15, so true.
  3. −2x + 3 > 15 at x = −4: −2 × (−4) + 3 = 11, not greater than 15. −4 is on the wrong side of −6, so false.
  4. −x + 3 > 15 at x = −12: −1 × (−12) + 3 = 15, which is not greater than 15. The circle is open, so the boundary is not a solution: false.

a > b means a − b > 0. From that definition come the rules: adding the same number to both sides keeps the direction; multiplying both sides by a positive number keeps it; multiplying or dividing by a negative number reverses it. Worked example 2 shows why.

A linear inequality in x and y is true on one side of its boundary line. The boundary is solid for ≤ and ≥ (points on it count) and dashed for < and > (they do not). To choose the side, test one point that is not on the line. Several inequalities together give the region where all are true.

9
2
1
28whole-number points in the region
nois the test point in the region

x ≥ 1, y ≥ 2 and x + y ≤ 9: the region is the triangle with corners (1, 2), (7, 2) and (1, 8), and it holds 28 whole-number points. The test point (2, 1) fails y ≥ 2 because 1 is less than 2, so it is outside.

Show me the steps
  1. k = 9 with the test point (2, 1): x ≥ 1 is true, y ≥ 2 is false because 1 is less than 2. The point is outside the region.
  2. The test point (3, 4): 3 ≥ 1, 4 ≥ 2 and 3 + 4 = 7 ≤ 9. All three are true, so it is inside.
  3. The test point (5, 4): 5 + 4 = 9, and 9 ≤ 9 is true, because ≤ includes the boundary. It is inside.
  4. k = 6: the line x + y = 6 moves in. The corners are (1, 2), (4, 2) and (1, 5), and the region holds 10 whole-number points.

Two common mistakes. Not reversing the sign when dividing by a negative number: −5x < 35 gives x > −7, not x < −7. And shading a side by feel: always test a point, because the side depends on the signs, not on which side looks “below”.

Worked examples

1. Solve 5 − 3x ≥ 20.

  1. Subtract 5 from both sides: −3x ≥ 15.
  2. Divide by −3 and reverse the sign: x ≤ −5. Check x = −6: 5 + 18 = 23 ≥ 20 is true.

2. Prove: if a > b and c < 0 then ac < bc.

  1. a > b means a − b > 0. Since c < 0, the product (a − b)c of a positive number and a negative number is negative.
  2. (a − b)c = ac − bc, so ac − bc < 0, which means ac < bc.

3. Describe the region y ≥ x − 2 and y < 7 − 2x.

  1. The lines are y = x − 2 (solid, because of ≥) and y = 7 − 2x (dashed, because of <). They meet where x − 2 = 7 − 2x, so x = 3 and y = 1.
  2. Test (0, 0): 0 ≥ −2 is true and 0 < 7 is true, so the region is the side containing the origin. The corner (3, 1) is not included, because 1 < 7 − 6 is false.

Your go. The first step is given. Solve the double inequality −2 < 3 − 5x ≤ 13. Subtract 3 from every part.

  1. −5 < −5x ≤ 10.
  2. Now divide every part by −5, reverse both signs, and write the answer with the smaller number first.
Check the answer

Dividing by −5 reverses both signs and swaps the ends: 1 > x ≥ −2, written −2 ≤ x < 1. Check x = 0: 3 − 0 = 3, between −2 and 13.

With a GDC (optional)

Many GDCs can shade the region of an inequality: enter the boundary as y = … and choose the shading option for above or below the line. Look at whether the boundary is drawn solid or dashed, because a calculator may not show that.

Whatever the screen shows, test one point by hand: put it in the inequality and see whether it is true. Everything on this page can be done without a GDC.

Check before you go on

1. Which is the solution of −6x > 54?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Solve −4x + 3 < 19. What is the smallest integer x that satisfies it?

3. How many integers x satisfy −3 < 2x + 1 ≤ 9?

Level 2 · Building

4. The boundary lines y = 2x − 1 and x + y = 8 meet at a corner of a region. What is the y-coordinate of the corner?

5. How many points with whole-number coordinates satisfy x ≥ 2, y ≥ 1 and 2x + y ≤ 12?

Level 3 · Stretch

6. The region y ≥ 0, y ≤ 2x and x + y ≤ 12 is a triangle. What is its area, in square units?

7. x is a number with −2 < x < 3, and y is a number with 1 < y < 4. What is the largest integer that x − y can equal?

Something to think about

In the second figure, the three boundary lines meet in a triangle. If the third inequality were changed to x + y ≥ 9 instead of x + y ≤ 9, what would the region look like, and would it still have a corner at (1, 2)?

One way to think about it

The region would be unbounded: it would run up and to the right with no third side, and it would have corners at (1, 8) and (7, 2), where the line x + y = 9 meets the other two. The point (1, 2) would no longer be in it, because 1 + 2 = 3 is not at least 9. An unbounded region has no largest value for most expressions, which matters in the next lesson.

Exit ticket

8. The line 3x − y = 6 is drawn dashed, and the shaded side contains the origin. Which inequality does the shading show?