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GCSE: Linear Inequalities
Skipton Girls High School
Objectives: Solving linear inequalities, combining inequalities and
representing solutions on number lines.
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
Means: x is (strictly) greater
? than 3.
0
1
2
3
4
x < -1
Means: x is (strictly) less?than -1.
5
-3
-2
-1
?
4
5
?
2
x≤5
Means: x is greater than?or equal to 4.
3
1
?
x≥4
2
0
6
7
Means: x is less than or equal
? to 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?
𝒙>𝟑
Can we add
or subtract to
both sides?
𝒙−𝟏>𝟐
Click to
 Deal
Click to 
No Deal
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
𝟐𝒙 > 𝟔
𝒙>𝟑
Click to
 Deal
Can we divide
both sides by
a positive
number?
Click to 
No Deal
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
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
‘Flipping’ the inequality
If we multiply or divide both sides of the inequality by a
negative number, the inequality ‘flips’!
OMG magic!
-2
2 < -4
4
Click to start
Bro-manimation
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
? 𝑥
−3 <
𝑥 > ?−3
1 − 3𝑥 ≥ 7
1 − 7 ?≥ 3𝑥
−6 ≥ ?3𝑥
−2 ≥ ?𝑥
?
𝑥 ≤ −2
Quickfire Examples
2𝑥 < 4
−𝑥 > −3
4𝑥 ≥ 12
−4𝑥 > 4
𝑥
− ≤1
2
Solve
Solve
Solve
Solve
Solve
𝑥 <? 2
𝑥 <? 3
𝑥 ≥? 3
𝑥 <?−1
𝑥 ≥?−2
Deal or No Deal?
We can manipulate inequalities in various ways, but which of these are allowed and
not allowed?
1
<2
𝑥
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!
More Examples
3𝑥 − 4 < 20
4𝑥 + 7 > 35
𝑥
5 + ≥ −2
2
7 − 3𝑥 > 4
𝑥
6− ≤1
3
Hint: Do the addition/subtraction before you do the
multiplication/division.
Solve
𝑥<
? 8
Solve
Solve
Solve
Solve
𝑥 >? 7
𝑥 ≥ ?−14
𝑥 <? 1
𝑥 ≥? 15
Dealing with multiple Hint:
inequalities
Do the addition/subtraction before you do the
multiplication/division.
8 < 5x
5x
-- 22 ≤ 23
and
2 < x and x ≤ 5
𝟐<𝒙≤𝟓
Click to start
bromanimation
More Examples
𝟏 < 𝟐𝒙 + 𝟑 < 𝟓
−𝟐 < −𝒙 < 𝟒
Hint: Do the addition/subtraction before you do the
multiplication/division.
Solve
Solve
−𝟏 < ?𝒙 < 𝟏
−𝟒 < ?𝒙 < 𝟐
Test Your Understanding
𝟏𝟏 < 𝟑𝒙 − 𝟒 < 𝟏𝟕
𝟏 < 𝟏 − 𝟐𝒙 < 𝟓
Solve
Solve
𝟓 < 𝒙? < 𝟕
−𝟐 < ?𝒙 < 𝟎
Exercise 1
Solve the following inequalities, and
illustrate each on a number line:
1
2
3
4
5
6
7
8
9
10
11
N1
2𝑥 − 1 > 5
𝒙 >?𝟑
−2𝑥 < 4
𝒙 >?−𝟐
5𝑥 − 2 ≤ 3𝑥 + 4
𝒙 ≤?𝟑
N2
𝑥
+1≥6
𝒙 ≥?𝟐𝟎
4
𝑦
−1≤7
𝒚 ≤?𝟒𝟖
6
1−𝑦
𝟏
≤𝑦
𝒚 ≥?
2
𝟑
1 − 4𝑥 > 5
𝒙 <?−𝟏
5 ≤ 2𝑥 − 1 < 9
𝟑 ≤ ?𝒙 < 𝟓
5 ≤ 1 − 2𝑥 < 9
− 𝟒 < ?𝒙 ≤ −𝟐
10 + 𝑥 < 4𝑥 + 1 < 33 𝟑 < 𝒙? < 𝟖
1 − 3𝑥 < 2 − 2𝑥 < 3 − 𝑥 𝒙 >?−𝟏
Sketch the graphs for
1
𝑦 = 𝑥 and 𝑦 = 1.
1
Hence solve 𝑥 > 1
0<x<1
?
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
>4
𝑥+2
If we assume 𝒙 + 𝟐 is positive, then 𝒙 >
𝟓
− 𝟐 and solving gives 𝒙 < − . Thus −𝟐 <
𝟒
𝟓
𝒙 < − as we had to assume 𝒙 > −𝟐. If
?
𝟒
𝟓
𝒙 < −𝟐 then this solves to 𝒙 > − which is
𝟒
a contradiction.
Thus −𝟐 < 𝒙 < −
𝟓
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”?
? x<4
x ≥ 2 and
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>3
x < -1 or
This is the only way you would
write this – you must use the
word ‘or’.
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
0
1
2
3
?
x < -1 or x > 4
4
5
-1
0
1
2
?
3
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
0
1
2
3
?
4
x < -1 or x > 4
5
-1
0
1
2
?
3
4
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
?
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
?
2
2<𝑥<5
𝒙>𝟓
5
4
𝑥<4
Combined
?𝟐 < 𝒙 < 𝟒
Test Your Understanding
?
1st
2nd
-1
condition
condition
Combined
-3
?
3
?
?
5
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
13
14
15
?
?
?
?
?
?
?
?
?
?
?
?1
?
?
2
9
?