Year 9 Inequalities

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 23rd March 2015

Objectives: Solving linear inequalities, combining inequalities and representing solutions on number lines.

Means: x is less than or equal to 4.

Writing inequalities and drawing number linesYou need to be able to sketch equalities and strict inequalities on a number line.

x > 3Means: x is (strictly) greater than 3.

0 1 2 3 4 5

?

This is known as a ‘strict’ inequality. x < -1

Means: x is (strictly) less than -1.

-3 -2 -1 0 1 2

?

x ≥ 4Means: x is greater than or equal to 4.

2 3 4 5 6 7

?x ≤ 5

2 3 4 5 6 7

?

? ?

? ?

Deal or No Deal?We can manipulate inequalities in various ways, but which of these are allowed and not allowed?

Click to Deal Click to No Deal

𝒙>𝟑 Can we add or subtract to both sides?

𝒙−𝟏>𝟐

Deal or No Deal?We can manipulate inequalities in various ways, but which of these are allowed and not allowed?

Click to Deal Click to No Deal

𝟐 𝒙>𝟔

𝒙>𝟑

Can we divide both sides by a positive number?

Deal or No Deal?We can manipulate inequalities in various ways, but which of these are allowed and not allowed?

Click to Deal Click to No Deal

𝒙<𝟏

𝟒 𝒙<𝟒

Can we multiply both sides by a positive number?

Deal or No Deal?We can manipulate inequalities in various ways, but which of these are allowed and not allowed?

Click to Deal Click to No Deal

𝒙<𝟏

−𝒙<−𝟏

Can we multiply both sides by a negative number?

2

If we multiply or divide both sides of the inequality by a negative number, the inequality ‘flips’!

< 4

Click to start Bro-manimation

-2 -4OMG magic!

‘Flipping’ the inequality

Alternative Approach

Or you could simply avoid dividing by a negative number at all by moving the variable to the side that is positive.

−𝑥<3 1−3 𝑥≥7?

?

?

?

?

?

Solve

Solve

Solve

Solve

Solve

Quickfire Examples

2 𝑥<4

−𝑥>−3

4 𝑥≥12

−4 𝑥>4−𝑥2≤1

𝑥<2

𝑥<3

𝑥≥3

𝑥<−1𝑥≥−2

?

?

?

?

?

Deal or No Deal?We can manipulate inequalities in various ways, but which of these are allowed and not allowed?

Click to Deal Click to No Deal

Can we multiply both sides by a variable?

The problem is, we don’t know if the variable has a positive or negative value, so negative solutions would flip it and positive ones wouldn’t. You won’t have to solve questions like this until Further Maths A Level!

1𝑥

<2

1<2 𝑥

Solve

Solve

Solve

Solve

Solve

Hint: Do the addition/subtraction before you do the multiplication/division.

3 𝑥−4<20

4 𝑥+7>35

5+𝑥2≥−2

7−3𝑥>4

6−𝑥3≤1

𝑥<8

𝑥>7

𝑥≥−14

𝑥<1

𝑥≥15

?

?

?

?

?

More Examples

8 < 5x - 2 ≤ 23

Hint: Do the addition/subtraction before you do the multiplication/division.

𝟐<𝒙 ≤𝟓

8 < 5x - 25x - 2 ≤ 23and

2 < x and x ≤ 52 < x x ≤ 5

Click to start bromanimation

Dealing with multiple inequalities

Solve−𝟏<𝒙<𝟏?

Hint: Do the addition/subtraction before you do the multiplication/division.

Solve−𝟒<𝒙<𝟐?

More Examples

𝟏<𝟐 𝒙+𝟑<𝟓

−𝟐<−𝒙<𝟒

Test Your Understanding

𝟏𝟏<𝟑𝒙 −𝟒<𝟏𝟕Solve

𝟓<𝒙<𝟕Solve

−𝟐<𝒙<𝟎

?

?𝟏<𝟏−𝟐 𝒙<𝟓

2 𝑥−1>5 𝒙>𝟑Exercise 1Solve the following inequalities, and illustrate each on a number line:

1

2

3

4

5

6

7

8

9

10

11

N1 Sketch the graphs for and . Hence solve 0 < x < 1

?

??

?

???

??

?

??N2

You can get around the problem of multiplying/dividing both sides by an expression involving a variable, by separately considering when it’s positive, and when it’s negative, and putting this together.Hence solve:

If we assume is positive, then and solving gives . Thus as we had to assume . If then this solves to which is a contradiction.Thus

?

Combining inequalitiesIt’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable.

x ≥ 2 and x < 4

AND How would we express “x is greater than or equal to 2, and less than 4”?

x ≥ 2, x < 4

2 ≤ x < 4

This last one emphasises the fact that x is between 2 and 4.

?

?

?

OR How would we express “x is less than -1, or greater than 3”?

x < -1 or x > 3?

This is the only way you would write this – you must use the word ‘or’.

Combining inequalitiesIt’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable.

2 ≤ x < 4 x < -1 or x > 4

0 1 2 3 4 5

?

-1 0 1 2 3 4

?

Combining inequalitiesIt’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable.

x ≥ 2 and x < 4 x < -1 or x > 4

0 1 2 3 4 5

?

-1 0 1 2 3 4

?

or and

To illustrate the difference, what happens when we switch them?

I will shoot you if I see any of these…

4>𝑥<8

4<𝑥>7

7 > > 4This is technically equivalent to:x > 7

This is technically equivalent to:x < 4

The least offensive of the three, but should be written:4 < x < 7

?

?

?

Combining Inequalities

2 5

4

2 5

4

2<𝑥<5

𝑥<4

Combined

Combined

𝒙>𝟓

𝟐<𝒙<𝟒

?

?

In general, we can combine inequalities either by common sense, or using number lines...

Where are you on both lines?

Test Your Understanding

-1 5

-3

Combined

3

?

?

?

?

1st condition

2nd condition

Exercise 2By sketching the number lines or otherwise, combine the following inequalities.

1234

56789

10

11

??

????

????

12

13

14

15

???

??