1. GCSE: Linear Inequalities
Skipton Girls High School
Objectives: Solving linear inequalities, combining inequalities and representing solutions on number lines.

2. Writing inequalities and drawing number lines
You need to be able to sketch equalities and strict inequalities on a number line.
This is known as a ‘strict’ inequality.
x > 3
x < -1
Means: x is (strictly) greater than 3. ?
Means: x is (strictly) less than -1. ? ? ?
x ≥ 4
x ≤ 5
Means: x is greater than or equal to 4. ?
Means: x is less than or equal to 5. ? ? ?

3. 𝒙>𝟑 𝒙−𝟏>𝟐 Deal or No Deal?  
We can manipulate inequalities in various ways, but which of these are allowed and not allowed?
𝒙>𝟑
Can we add or subtract to both sides?
𝒙−𝟏>𝟐
Click to Deal  
Click to No Deal

4. 𝟐𝒙>𝟔 𝒙>𝟑 Deal or No Deal?  
We can manipulate inequalities in various ways, but which of these are allowed and not allowed?
𝟐𝒙>𝟔
Can we divide both sides by a positive number?
𝒙>𝟑
Click to Deal  
Click to No Deal

5. 𝒙<𝟏 𝟒𝒙<𝟒 Deal or No Deal?  
We can manipulate inequalities in various ways, but which of these are allowed and not allowed?
𝒙<𝟏
Can we multiply both sides by a positive number?
𝟒𝒙<𝟒
Click to Deal  
Click to No Deal

6. 𝒙<𝟏 −𝒙<−𝟏 Deal or No Deal?  
We can manipulate inequalities in various ways, but which of these are allowed and not allowed?
𝒙<𝟏
Can we multiply both sides by a negative number?
−𝒙<−𝟏
Click to Deal  
Click to No Deal

7. -4 -2 2 < 4 ‘Flipping’ the inequality × (-1) × (-1)
If we multiply or divide both sides of the inequality by a negative number, the inequality ‘flips’!
OMG magic! -4 -2 2
< 4
× (-1)
Click to start
Bro-manimation
× (-1)

8. −𝑥<3 −3<𝑥 𝑥>−3 1−3𝑥≥7 1−7≥3𝑥 −6≥3𝑥 −2≥𝑥 𝑥≤−2
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 −3<𝑥 𝑥>−3
1−3𝑥≥7 1−7≥3𝑥 −6≥3𝑥 −2≥𝑥 𝑥≤−2 ? ? ? ? ? ?

9. 2𝑥<4 𝑥<2 𝑥<3 −𝑥>−3 4𝑥≥12 𝑥≥3 −4𝑥>4 𝑥<−1 − 𝑥 2 ≤1
Quickfire Examples
Solve
2𝑥<4
𝑥<2 ?
Solve
𝑥<3
−𝑥>−3 ?
Solve
4𝑥≥12
𝑥≥3 ?
Solve
−4𝑥>4
𝑥<−1 ?
− 𝑥 2 ≤1
Solve
𝑥≥−2 ?

10. 1 𝑥 <2 1<2𝑥 Deal or No Deal?  
We can manipulate inequalities in various ways, but which of these are allowed and not allowed?
1 𝑥 <2
Can we multiply both sides by a variable?
1<2𝑥
Click to Deal 
Click to No Deal 
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!

11. 3𝑥−4<20 𝑥<8 4𝑥+7>35 𝑥>7 5+ 𝑥 2 ≥−2 𝑥≥−14 7−3𝑥>4 𝑥<1
More Examples
Hint: Do the addition/subtraction before you do the multiplication/division.
Solve
3𝑥−4<20
𝑥<8 ?
Solve
4𝑥+7>35
𝑥>7 ?
Solve
5+ 𝑥 2 ≥−2
𝑥≥−14 ?
Solve
7−3𝑥>4
𝑥<1 ?
Solve
6− 𝑥 3 ≤1
𝑥≥15 ?

12. Click to start bromanimation
Dealing with multiple inequalities
Hint: Do the addition/subtraction before you do the multiplication/division.
8 < 5x - 2 ≤ 23
8 < 5x - 2
5x - 2 ≤ 23
and
2 < x and x ≤ 5
2 < x
x ≤ 5
𝟐<𝒙≤𝟓
Click to start bromanimation

13. −𝟏<𝒙<𝟏 −𝟒<𝒙<𝟐
More Examples
Hint: Do the addition/subtraction before you do the multiplication/division.
Solve
𝟏<𝟐𝒙+𝟑<𝟓
−𝟏<𝒙<𝟏 ?
Solve
−𝟐<−𝒙<𝟒 ?
−𝟒<𝒙<𝟐

14. 𝟓<𝒙<𝟕 −𝟐<𝒙<𝟎 𝟏𝟏<𝟑𝒙−𝟒<𝟏𝟕 𝟏<𝟏−𝟐𝒙<𝟓
Test Your Understanding
Solve
𝟓<𝒙<𝟕
𝟏𝟏<𝟑𝒙−𝟒<𝟏𝟕 ?
Solve
𝟏<𝟏−𝟐𝒙<𝟓
−𝟐<𝒙<𝟎 ?

15. Exercise 1
Solve the following inequalities, and illustrate each on a number line:
Sketch the graphs for 𝑦= 1 𝑥 and 𝑦=1. Hence solve 1 𝑥 >1
0 < x < 1 N1
2𝑥−1> 𝒙>𝟑 −2𝑥< 𝒙>−𝟐 5𝑥−2≤3𝑥 𝒙≤𝟑 𝑥 4 +1≥ 𝒙≥𝟐𝟎 𝑦 6 −1≤ 𝒚≤𝟒𝟖 1−𝑦 2 ≤𝑦 𝒚≥ 𝟏 𝟑 1−4𝑥> 𝒙<−𝟏 5≤2𝑥−1< 𝟑≤𝒙<𝟓 5≤1−2𝑥< −𝟒<𝒙≤−𝟐 10+𝑥<4𝑥+1< 𝟑<𝒙<𝟖 1−3𝑥<2−2𝑥<3−𝑥 𝒙>−𝟏 1 ? 2 ? ? 3 ?
You can get around the problem of multiplying/dividing both sides by an expression involving a variable, by separately considering when the denominator positive, and when it’s negative, and putting this together.
Hence solve:
3 𝑥+2 >4
If we assume 𝒙+𝟐 is positive, then 𝒙>−𝟐 and solving gives 𝒙<− 𝟓 𝟒 . Thus −𝟐<𝒙<− 𝟓 𝟒 as we had to assume 𝒙>−𝟐. If 𝒙<−𝟐 then this solves to 𝒙>− 𝟓 𝟒 which is a contradiction.
Thus −𝟐<𝒙<− 𝟓 𝟒 N2 4 ? 5 ? ? 6 7 ? 8 ? ? 9 ? 10 ? 11 ?

16. x ≥ 2 and x < 4 x < -1 or x > 3 x ≥ 2, x < 4 2 ≤ x < 4
Combining inequalities
It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable.
AND
How would we express
“x is greater than or equal to 2, and less than 4”? OR
How would we express
“x is less than -1, or greater than 3”?
x ≥ 2 and x < 4
x < -1 or x > 3 ? ?
x ≥ 2, x < 4 ?
This is the only way you would write this – you must use the word ‘or’.
2 ≤ x < 4 ?
This last one emphasises the fact that x is between 2 and 4.

17. Combining inequalities
It’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 ? ?

18. x ≥ 2 and x < 4 x < -1 or x > 4
Combining inequalities
It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable.
To illustrate the difference, what happens when we switch them? or
and
x ≥ 2 and x < 4
x < -1 or x > 4 ? ?

19. 7 > 𝑥 > 4 4>𝑥<8 4<𝑥>7
I will shoot you if I see any of these…
4>𝑥<8
This is technically equivalent to:
x < 4 ?
4<𝑥>7
This is technically equivalent to:
x > 7 ?
7 > 𝑥 > 4
The least offensive of the three, but should be written:
4 < x < 7 ?

20. Combining Inequalities
In general, we can combine inequalities either by common sense, or using number lines... 2 5
Where are you on both lines? 4
Combined ?
𝒙>𝟓 2 5
2<𝑥<5 4
𝑥<4 ?
Combined
𝟐<𝒙<𝟒

21. Test Your Understanding? ? -1 5
1st condition ? -3 3
2nd condition ?
Combined

22. Exercise 2
By sketching the number lines or otherwise, combine the following inequalities. 1 ? 2 ? 3 ? 4 ? ? 5 6 ? 7 ? 8 ? 9 ? 10 ? ? 11 12 ? 1 13 ? 14 ? 15 ?
2 9