15
Inequalities
CHAPTER
In this chapter the inequality signs, , , and are used.
means less than, for example 3 4
means greater than , for example 5 4
means less than or equal to
means greater than or equal to.
15.1 Inequalities on a number line
The inequality x 1 is shown on a number line.
The open circle shows that x 1 is not included.
5 4 3 2 1
0
1
2
3
4
5
The inequality x 4 is shown on a number line.
The filled circle shows that x 4 is included.
5 4 3 2 1
0
1
2
3
4
5
This number line shows the values of x which satisfy
both x 1 and x 4
We write this as 1 x 4
5 4 3 2 1
0
1
2
3
4
5
5 4 3 2 1
0
1
2
3
4
5
Example 1
Write down the inequality shown on the number line.
Solution 1
x3
The open circle indicates that 3 is not included.
Example 2
Show the inequality x 2 on a number line.
Solution 2
5 4 3 2 1
0
1
2
3
4
The filled circle indicates that 2 is included.
5
Exercise 15A
1 Write down the inequality shown on the number line:
a
b
5 4 3 2 1
0
1
2
3
4
5
c
0
1
2
3
4
5
5 4 3 2 1
0
1
2
3
4
5
d
5 4 3 2 1
240
5 4 3 2 1
0
1
2
3
4
5
15.2 Solving inequalities
CHAPTER 15
e
f
5 4 3 2 1
0
1
2
3
4
5
5 4 3 2 1
0
1
2
3
4
5
5 4 3 2 1
0
1
2
3
4
5
g
Answer questions 2 to 4 on the resource sheet.
2 Show these inequalities on a number line.
b x5
a x2
c x0
d x 1
3 Show on a number line the values of x which satisfy both
b x 1 and x 0
a x 1 and x 3
c x 4 and x 1
4 Show these inequalities on a number line.
b 4 x 0
a 1 x 3
c 5 x 2
15.2 Solving inequalities
Inequalities can be solved in a similar way to equations but the inequality sign must be kept throughout
and the solution is an inequality with the letter on its own.
Example 3
Solve the inequality 2x 1 7
Solution 3
2x 1 7
2x 1 1 7 1
2x 8
Add 1 to both sides.
Divide both sides by 2
x 4
The solution is an inequality
with the letter on its own.
Example 4
a Solve the inequality 5x x 10
b Show your solution on a number line.
Solution 4
a
5x x 10
5x x x 10 x
4x 10x
x 2.5
Subtract x from both sides.
Divide both sides by 4
b
5
4
3
2
1
0
1
2 2.5 3
4
5
The inequality 2 1 is true.
Multiplying both sides by 3 gives 6 3 which is not true but 6 3 is true.
In general, multiplying or dividing both sides of an inequality by a negative number changes the
direction of the inequality.
To avoid this problem rearrange the terms so that the final term in x is positive as in the next example.
241
Inequalities
CHAPTER 15
Example 5
Example 6
Solve the inequality 2x 6
x
Solve the inequality 2 1 1
3
Solution 5
x
2 1 1
3
2x
2 1
Expand the bracket.
3
Solution 6
When solving 2x 6 the common error is to divide
both sides by 2 to get x 3
This is incorrect since x 1 satisfies x 3 but when
x 1, 2x 2 which is not greater than 6
2x
1
3
2x 3
x 1.5
2x 6
Subtract 2 from both sides.
Multiply both sides by 3
2x 2x 6 2x
Rearrange the terms so that
the x term is positive.
Add 2x to both sides.
0 6 2x
Divide both sides by 2
6 2x
Subtract 6 from both sides.
3 x
Divide both sides by 2
x 3
3 is greater than x means
the same as x is less than 3
Example 7
Solve the inequality 2(x 1) 7(x 2)
Solution 7
2(x 1) 7(x 2)
2x 2 7x 14
2 14 7x 2x
Expand the brackets.
Rearrange the terms so that the final term in x is positive.
16 5x
3.2 x
Divide both sides by 5
x 3.2
Exercise 15B
1 Solve these inequalities.
b x27
a x15
2 Solve these inequalities.
b 3x 1 8
a 2x 1 9
c 2x 8
d 3x 6
e 4x 3
c 4x 3 15
d 5x 12 2
e 6x 3 0
d 7x 2x 35
e 9x 5x 18
3 a Solve the inequality 10x 1 16
b Show your solution on a number line.
4 Solve these inequalities.
a 3x 2x 10 b 5x x 12
242
c 4x x 6
15.3 Integer solutions to inequalities
5 a Solve the inequality 3x 2 x
CHAPTER 15
b Show your solution on a number line.
6 Solve the inequality 5x 7 19
7 Solve these inequalities.
x1
b 4
3
x
a 4
5
2x 1
d 2 3
5x
c 2
8
3x
e x 2
4
8 Solve these inequalities.
a 4(x 1) 7
d 1 7(x 2) 5x 3
b 5x 2(x 6)
e 5(x 3) 2 3(x 4)
c 9(x 2) 5(x 2)
9 Solve these inequalities.
a 4x 8
d 1 2(3x 2) 5(2x 3)
b 2 5x 8 2x
e 7 4(x 3) 2x
c 2(4 5x) 19
4x
a 3x 2 7
5
x
b 1 2(x 1)
2
x
c 4 3(x 1)
2
x2 x1
d 3
2
2 5x x
e 3
4
2
10 Solve these inequalities.
15.3 Integer solutions to inequalities
The inequality 4 x 2 is shown on the number line.
5 4 3 2 1
0
1
2
3
4
5
Integers are whole numbers. They can be positive or negative or zero.
So the integer values which satisfy the inequality 4 x 2 are 3, 2, 1, 0, 1, 2
Example 8
4 2n 3
n is an integer. Find all the possible values of n.
Solution 8
4 2n 3
4 2n
AND
2 n
AND
2n 3
n 1.5
Split 4 2n 3
Solve both inequalities.
2 n 1.5
Combine the two inequalities.
The possible values of n are 1, 0, 1
Write down the integer values satisfying 2 n 1.5
Example 9
4 2p 1 12
p is a positive integer. Find all the possible values of p.
243
Inequalities
CHAPTER 15
Solution 9
4 2p 1 12
4 2p 1 AND
2p 1 12
Split 4 2p 1 12
5 2p
2.5 p
2p 11
p 5.5
Solve both inequalities.
AND
AND
2.5 p 5.5
Combine the two inequalities.
The possible values of p are 1, 2, 3, 4, 5
Write down the positive integer values satisfying 2.5 p 5.5
Note 0 is not a positive integer.
15.4 Problems involving inequalities
Example 10
A rectangular field has length (3x 5) metres and width (2x 3) metres.
a Explain why x 1.5
b The perimeter of the field is no more than 60 metres. Find the greatest possible length of the field.
Solution 10
a 2x 3 0 since the width must be positive 2x 3 so x 1.5
b Perimeter 2(3x 5) 2(2x 3)
2(3x 5) 2(2x 3) 60
6x 10 4x 6 60
10x 4 60
10x 56
x 5.6
3x 5 3 5.6 5
Length 21.8
Perimeter 2l 2w
Perimeter 60 (no more than 60 so it can be equal to 60).
Solve the inequality.
x 5.6 so 3x 3 5.6 and 3x 5 3 5.6 5
The greatest possible length is 21.8 metres
Exercise 15C
1 a Show the inequality 2 y 4 on a number line.
b If y is an integer, use your number line to write down all the possible values of y.
2 a Show the inequality 3 p 2 on a number line.
b If p is an integer, use your number line to write down all the possible values of p.
3 8 y 5
y is an integer.
Write down all the possible values of y.
4 a Solve 6 2x 4
b Given that x is an integer and 6 2x 4, write down all the possible values of x.
5 a Solve 3 4w 13
b Given that w is an integer and 3 4w 13, write down all the possible values of w.
244
15.5 Solving inequalities graphically
CHAPTER 15
6 a Solve 3 x 1 5
b Given that x is an integer and 3 x 1 5, write down all the possible values of x.
7 2p 9
p is a positive integer.
Find all the possible values of p.
8 5q 1 19
q is a positive integer.
Find all the possible values of q.
9 2x 10 1
x is a negative integer.
Find all the possible values of x.
10 a Solve 7 2p 3 9
b p is a positive integer and 7 2p 3 9
Find all the possible values of p.
2x
11 a Solve 1 2
3
b 4 5(x 1) 10
c 3 2(1 2x) 6
12 A rectangular field has length (2x 8) metres and width (4x 3) metres.
The perimeter of the field is no more than 112 metres. Find the greatest possible area of the field.
13 The Smiths have four daughters. The table shows some information about their ages in years.
Ages in years
Ann
Betty
Cath
Debra
x
2x 3
12
20
Betty is older than Cath but younger than Debra.
Work out all Ann’s possible ages.
15.5 Solving inequalities graphically
y
3
2 y1
y1
3
2
1
1
O
1
2
3 x
1
y1
2
3
The diagram shows the line with equation y 1
The coordinates of all points on this line satisfy the equation y 1
All points above this line have a y-coordinate greater than 1 and so satisfy the inequality y > 1
All points below this line have a y-coordinate less than 1 and so satisfy the inequality y < 1
245
Inequalities
CHAPTER 15
y
y
y1
y1
y1
O
A solid line as a boundary is considered
to be part of the shaded region.
A dashed boundary is not part of the
shaded region.
y1
x
x
O
Each diagram above shows a region shaded above the line y 1
When the line is solid, the coordinates of all points in the shaded region satisfy the inequality y 1
When the line is dashed, the coordinates of all points in the shaded region satisfy the inequality y 1
The diagram shows a shaded region bounded by the lines x 1 and x 2
x 1
y
x2
4
2
4
O
2
2
4 x
2
4
The coordinates of all points in this region satisfy both the inequalities
x 1 (since this boundary is dashed)
x 2 (since this boundary is solid)
This is written as 1 x 2
The diagram shows the shaded region bounded by the lines x 1, x 2, y 1 and y 4
x 1
x2
y
6
y4
4
2
y1
2
1
O
1
2
3 x
2
4
The coordinates of all points in this region satisfy both 1 x 2 and 1 y 4
246
15.5 Solving inequalities graphically
CHAPTER 15
Example 11
On a grid, shade the region of points whose coordinates satisfy the inequalities
3 x 0 and 2 y 1
Solution 11
x 3
x0
y
6
4
2
4
y1
O
2
2
4
A dashed line is drawn very close
to the y-axis to show that points
on this boundary are not included.
3 x
y 2
2
4
The diagram shows the line with
equation x y 3
The coordinates of all points on this
line satisfy the equation x y 3
The coordinates of all points above this
line satisfy the inequality x y > 3
The coordinates of all points below this
line satisfy the inequality x y < 3
y
6
For example, the
point (4, 2) lies
above the line
x y 3 and
since 4 2 6, its
coordinates satisfy
the inequality
xy3
5
4
xy3
3
xy3
2
1
xy3
2 1 O
1
1
2
3
4
5
6 x
2
Example 12
y
a Write down the three inequalities which must be satisfied
by the coordinates of all points in the shaded region.
b x and y are integers. Mark with a cross () the three points
which satisfy the three inequalities.
5
xy3
4
3
2
Solution 14
a
y 1
The shaded region is to the right of the solid y-axis (x 0)
xy 3
y1
1
The shaded region is above the solid line y = 1
x 0
b
6
3 2 1 O
1
The shaded region is below the dashed line x y 3
1
2
3
4
5
6 x
2
y
4
3
2
1
1 0
1
2
3 x
The three points must be in the shaded region
which includes the lines x 0 and y = 1
The coordinates of the three points are
(0, 1), (0, 2) and (1, 1).
247
Inequalities
CHAPTER 15
Example 13
a
i On a grid scaled from 0 to 8 on each axis, draw the line with equation x y 5
ii On the same grid draw the line with equation 2y x 4
b On the grid, shade the region of points whose coordinates satisfy the three inequalities x y 5,
2y x 4, 2x 7
c From your graph, write down the solution of the simultaneous equations
xy 5
2y x 4
d Write down the least value of x in the shaded region.
Solution 13
a b y
8
2x 7
7
6
5 xy5
2y x 4
4
3
2
1
O
1
2
3
4
5
6
7
8 x
a i x y 5 passes through (0, 5) and (5, 0)
ii To draw the line 2y x 4 plot and join some of the points
(0, 2) (2, 3) (4, 4) (6, 5)
Or write the equation in the form y 0.5x 2 and draw
the line with gradient 0.5 and intercept 2 on the y-axis.
b Draw the dashed vertical line 2x 7 or x 3.5 for the
inequality 2x 7.
The shaded region will be:
● to the left of the line 2x 7 (since 2x 7)
● above the solid line x y 5 (since x y 5)
● below the solid line 2y x 4 (since 2y x 4)
c x 2, y 3
Find the coordinates of the point where the line x y 5 and
the line 2y x 4 intersect.
d Least value of x is 2
Find the x-coordinate of the point which is furthest to the left in
the shaded region.
Exercise 15D
1 For each of these questions use the resource sheet with a grid scaled from 5 to 5 on each
axis to shade the region of points whose coordinates satisfy
b y 1
c 0x2
d 4 y 4
a x3
f 1 x 5 and 0 y 3
e 1 x 3 and 2 y 0
h 3 x 1 and 3 y 1
g 2 x 3 and 4 y 1
j yx
k y x
i 0 x 3 and 2 y 0
l y x 1
m xy4
o 2x y 4, y x 1 and y 0
n x y 1 and x y 3
2 The diagram shows a shaded region bounded by three lines.
a Write down the equation of each of the three lines.
b Write down the three inequalities satisfied by the
coordinates of the points in the shaded region.
c Write down the greatest value of x in the shaded region.
y
4
3
2
1
2 1 O
1
2
248
1
2
3
4
5
6 x
15.5 Solving inequalities graphically
CHAPTER 15
3 The diagram shows the line with equation y 2x 3
a On a copy of the grid, draw the line 2x y 6
b Shade the region for which y 2x 3, 2x y 6 and
x1
c x and y are integers. Write down the coordinates of the
two points which satisfy the three inequalities.
y
4
y 2x 3
3
2
1
1
O
1
1
2
3 x
2
3
4
4 a Show that the points (0, 4) and (6, 0) lie on the line with equation 3y 2x 12
b The diagram shows a shaded region.
y
6
5
4
3
2
1
O
1
2
3
4
5
6 x
Write down the three inequalities which must be satisfied by the coordinates of all points in the
shaded region.
c x and y are integers. Write down the coordinates of the two points which satisfy the three
inequalities.
5 The diagram shows the lines with equations y 2x and y x 3
y
8
7
6
5
yx3
4
3
y 2x
2
1
2
1
O
1
1
2
3 x
2
3
a From the graph, write down the solution of the simultaneous equations
y 2x
yx3
b Shade the region for which x y 1, y x 3, y 2x and 2y 5
c x and y are integers. Write down the coordinates of the set of points which satisfy all four
inequalities, x y 1, y x 3, y 2x and 2y 5
249
Inequalities
CHAPTER 15
6 a 6x 5y 30
x and y are both integers.
Write down two possible pairs of values, (x …, y …),
that satisfy this inequality.
b 6x 5y 30, 2x y, y 1, x 0
x and y are both integers.
On the resource sheet, mark with a cross (), each
of the three points which satisfy all these four inequalities.
y
8
6
4
2
2
O
2
4
6 x
2
Chapter summary
You should now know that:
the signs used for inequalities are , , and to show an inequality on a number line, you use a together with a line for and and
use a together with a line for and solving inequalities is similar to solving equations but the inequality sign must be kept throughout
and the solution is an inequality with a letter of its own
multiplying or dividing both sides of an inequality by a negative number will change the
direction of an inequality
to find integer solutions to inequalities, a number line can be used, for example, the integer
solutions to the inequality 2 p 1 are 1, 0 and 1
to solve inequalities graphically, draw regions bounded by straight lines
a solid line as a boundary is considered to be part of the shaded region, a dashed
boundary is not part of the shaded region.
means less than
means greater than
means less than or equal to
means greater than or equal to
Chapter 15 review questions
1 6 y 3
y is an integer.
Write down all the possible values of y.
2 Solve 5x 3 19
3 a Solve the equation 5 3x 2(x 1)
b 3 y 3
y is an integer.
Write down all the possible values of y.
250
(1388 March 2005)
(1388 June 2005)
Chapter 15 review questions
CHAPTER 15
4 5p 17
p is a positive integer.
Write down all the possible values of p.
5 a Solve the inequality 3x 7 x
b Show your answer to part a on a number line.
6 5 2m 9
m is an integer.
Write down all the possible values of m.
7 3x 20 3
x is a negative integer.
Write down all the possible values of x.
8 2 6y 19
y is a negative integer.
Write down all the possible values of y.
9 Solve the inequality 3x 2 7
(1388 June 2003)
10 Solve 3x 4 16
(1388 Mock)
11 a Solve the inequality 5x 12 2
b Expand and simplify (x 6)(x 4).
12 Solve 4 x 2 7
(1388 January 2003)
(1388 March 2003)
13 n is a whole number such that 7 3n 15
List all the possible values of n.
14 a Solve the equation 4x 3 2(x 3)
(1388 January 2004)
b Solve the inequality 2x 3 8
15 n is a whole number such that 6 2n 13
List all the possible values of n.
(1385 June 1999)
(1385 November 2000)
16 a Solve the equation 7(x 1) 2x 1
b i Solve the inequality 4y 3 1
ii Write down the smallest integer value of y which satisfies the inequality 4y 3 1
(1385 June 2002)
17 n is an integer such that 5 2n 6
a List all the possible values of n.
b Solve the inequality 5 x 5x 11
(1387 Mock)
18 a Solve the inequality 4x 3 7
An inequality is shown on the number line.
4 3 2 1
0
b Write down the inequality.
1
2
3
4
(1388 March 2004)
251
Inequalities
CHAPTER 15
19 a Solve the inequality 7x 3 17
x is a whole number such that 7x 3 17
b Write down the smallest value of x.
(1388 November 2005)
20 a 4x 3y 12
x and y are both integers.
Write down two possible pairs of values, (x …, y …), that satisfy this inequality.
y 3x, y 0, x 0
b 4x 3y 12,
x and y are both integers.
y
5
4
3
2
1
5 4 3 2 1 O
1
1
2
3
4
5 x
2
3
4
5
On the resource sheet, mark with a cross () each of the three points which satisfy
(1387 November 2005)
all these four inequalities.
21 a On the resource sheet draw the line with equation x 2y 6
y
10
8
6
4
2
O
2
4
6
8
10 x
b Shade the region for which x 2y 6, 0 x 4 and y 0.
(1385 November 1997)
22 The perimeter of this rectangle has to be more than 11 cm and less than 20 cm.
Diagram NOT
accurately drawn
3 cm
x cm
a Show that 5 2x 14
b x is an integer. List all the possible values of x.
252
(1385 November 2002)
Chapter 15 review questions
CHAPTER 15
23 a 2 x 1, x is an integer.
Write down all the possible values of x.
b 2 x 1, y 2, y x 1,
x and y are integers.
On the resource sheet, mark with a cross () each
of the six points which satisfies all these three
(1387 June 2003)
inequalities.
y
5
4
3
2
1
5 4 3 2 1 O
1
1
2
3
4
5 x
4
5
6
7
8 x
2
3
4
5
24 a On the resource sheet, draw straight lines and use
shading to show the region R that satisfies the
inequalities x 1, y x 1 and x y 7
b Write down the coordinates of all the points of
R whose coordinates are both integers.
y
8
7
6
5
4
3
2
1
O
1
2
3
253