Sec 3 E-Math Inequalities worksheet with answers

A free Sec 3 E-Math worksheet on inequalities: 9 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 3 E-Math worksheet: Inequalities

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

Solve

-5x + 6 < -14

and

-3x - 5 > -29

Answer[2]

(a)Solve

-9 - 5x \leq 6

Answer (a)[1]

(b)Solve

\dfrac{3x + 5}{3} \geq \dfrac{3x + 3}{2}

Answer (b)[2]

(a)

x

and

y

are integers such that

-1 < x < 6

and

-6 < y < -2

. Find the largest possible value of

x^2 + y^2

Answer (a)[2]

(b)

x

and

y

are integers such that

-5 < x \leq 1

and

-1 < y \leq 6

. Find the smallest possible value of

y^2 - x^2

Answer (b)[2]

Section B

The number line represents the values of

x

that satisfy an inequality.

(a)Write down the inequality shown.

Answer (a)[1]

(b)Write down the largest integer value of

x

shown.

Answer (b)[1]

Solve the pair of simultaneous inequalities

5x + 7 \geq 47

and

5x - 8 \leq 2

, or explain why there is no solution.

Answer[3]

A taxi charges a flag-down fare of $3.20, plus $0.60 for every kilometre travelled. Ali has $19 and takes the taxi for

x

km.

(a)Write down an inequality in

x

[1]

(b)Solve your inequality, and hence find the greatest number of whole kilometres Ali can travel.

Answer (b)km[2]

Three consecutive odd numbers are

n

n + 2

and

n + 4

. Their sum is more than

80

and less than

100

(a)Form a pair of inequalities in

n

and solve them.

Answer (a)[3]

(b)Hence write down all the possible values of

n

Answer (b)[1]

Mei buys

x

pens at $1.20 each and

(x + 3)

notebooks at $2.50 each. She spends at most $40, and she buys at least

5

pens.

(a)Show that

37x \leq 325

[2]

(b)Find all the possible numbers of pens Mei could buy.

Answer (b)[2]

A rectangle is

(2x + 1)

cm long and

5

cm wide. Its perimeter is at most

40

cm and its area is at least

30

cm².

A rectangle labelled (2x + 1) cm long and 5 cm wide

(a)Show that

x \leq 7

[2]

(b)Use the area to form another inequality, and hence find the range of values of

x

Answer (b)[2]

(c)Find the largest possible area of the rectangle.

Answer (c)cm²[1]

END OF PAPER

Answers and worked solutions

Question 1

Answer:

4 < x < 8

[2]

-5x + 6 < -14

gives

x > 4

-3x - 5 > -29

gives

x < 8

Yes: they overlap.

4 < x < 8

Question 2

(a)Answer:

x \geq -3

(dividing by a negative number reverses the sign)[1]

-5x \leq 15

x \geq -3

(dividing by a negative number reverses the sign)

(b)Answer:

x \leq \dfrac{1}{3}

(dividing by a negative number reverses the sign)[2]

2(3x + 5) \geq 3(3x + 3)

6x + 10 \geq 9x + 9

-3x \geq -1

x \leq \dfrac{1}{3}

(dividing by a negative number reverses the sign)

Question 3

(a)Answer:

25 + 25 = 50

[2]

x

0

1

2

3

4

5

y

-5

-4

-3

y^2 = 25

x^2 = 25

25 + 25 = 50

(b)Answer:

0 - 16 = -16

[2]

x

-4

-3

-2

-1

0

1

y

0

1

2

3

4

5

6

y^2 = 0

x^2 = 16

0 - 16 = -16

Question 4

(a)Answer:

x \leq -2

[1]

A filled circle means the value is included (

\leq

\geq

); an open circle means it is not (

<

>

x \leq -2

(b)Answer:

-2

[1]

x \leq -2

Answer:

-2

Question 5

Answer:

\text{no solution}

[3]

5x + 7 \geq 47

gives

x \geq 8

5x - 8 \leq 2

gives

x \leq 2

.The two ranges point away from each other and never overlap, so there is no solution.

Question 6

(a)[1]

Total fare

= 3.20 + 0.60x

, and it must be at most $19:

3.20 + 0.60x \leq 19

(b)Answer:

26

km[2]

0.60x \leq 15.80

x \leq 26.33\ldots

The greatest whole number of kilometres is

26

Question 7

(a)Answer:

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

[3]

n + (n + 2) + (n + 4) = 3n + 6

80 < 3n + 6 < 100

74 < 3n < 94

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

(b)Answer:

25, 27, 29, 31

[1]

n

is odd and

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

n = 25, 27, 29

31

Question 8

(a)[2]

1.20x + 2.50(x + 3) \leq 40

3.7x + 7.5 \leq 40

3.7x \leq 32.5

37x \leq 325

(shown)

(b)Answer:

5, 6, 7, 8

[2]

x \leq \dfrac{325}{37} = 8.78\ldots

x \geq 5

, and

x

is a whole number

x = 5, 6, 7

8

Question 9

(a)[2]

2(2x + 1 + 5) \leq 40

4x + 12 \leq 40

x \leq 7

(shown)

(b)Answer:

2.5 \leq x \leq 7

[2]

5(2x + 1) \geq 30

2x + 1 \geq 6

x \geq 2.5

Together with part (a):

2.5 \leq x \leq 7

(c)Answer:

75

cm²[1]

The area is largest when

x = 7

5(2 \times 7 + 1) = 75

cm²

Practice questions by skill

Sec 3 E-Math Inequalities worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Inequalities notes. Or practise it online with step-by-step help when you get one wrong.

Answers and worked solutions

Question 1

Answer:

4 < x < 8

[2]

-5x + 6 < -14

gives

x > 4

-3x - 5 > -29

gives

x < 8

Yes: they overlap.

4 < x < 8

Question 2

(a)Answer:

x \geq -3

(dividing by a negative number reverses the sign)[1]

-5x \leq 15

x \geq -3

(dividing by a negative number reverses the sign)

(b)Answer:

x \leq \dfrac{1}{3}

(dividing by a negative number reverses the sign)[2]

2(3x + 5) \geq 3(3x + 3)

6x + 10 \geq 9x + 9

-3x \geq -1

x \leq \dfrac{1}{3}

(dividing by a negative number reverses the sign)

Question 3

(a)Answer:

25 + 25 = 50

[2]

x

0

1

2

3

4

5

y

-5

-4

-3

y^2 = 25

x^2 = 25

25 + 25 = 50

(b)Answer:

0 - 16 = -16

[2]

x

-4

-3

-2

-1

0

1

y

0

1

2

3

4

5

6

y^2 = 0

x^2 = 16

0 - 16 = -16

Question 4

(a)Answer:

x \leq -2

[1]

A filled circle means the value is included (

\leq

\geq

); an open circle means it is not (

<

>

x \leq -2

(b)Answer:

-2

[1]

x \leq -2

Answer:

-2

Question 5

Answer:

\text{no solution}

[3]

5x + 7 \geq 47

gives

x \geq 8

5x - 8 \leq 2

gives

x \leq 2

.The two ranges point away from each other and never overlap, so there is no solution.

Question 6

(a)[1]

Total fare

= 3.20 + 0.60x

, and it must be at most $19:

3.20 + 0.60x \leq 19

(b)Answer:

26

km[2]

0.60x \leq 15.80

x \leq 26.33\ldots

The greatest whole number of kilometres is

26

Question 7

(a)Answer:

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

[3]

n + (n + 2) + (n + 4) = 3n + 6

80 < 3n + 6 < 100

74 < 3n < 94

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

(b)Answer:

25, 27, 29, 31

[1]

n

is odd and

24\tfrac{2}{3} < n < 31\tfrac{1}{3}

n = 25, 27, 29

31

Question 8

(a)[2]

1.20x + 2.50(x + 3) \leq 40

3.7x + 7.5 \leq 40

3.7x \leq 32.5

37x \leq 325

(shown)

(b)Answer:

5, 6, 7, 8

[2]

x \leq \dfrac{325}{37} = 8.78\ldots

x \geq 5

, and

x

is a whole number

x = 5, 6, 7

8

Question 9

(a)[2]

2(2x + 1 + 5) \leq 40

4x + 12 \leq 40

x \leq 7

(shown)

(b)Answer:

2.5 \leq x \leq 7

[2]

5(2x + 1) \geq 30

2x + 1 \geq 6

x \geq 2.5

Together with part (a):

2.5 \leq x \leq 7

(c)Answer:

75

cm²[1]

The area is largest when

x = 7

5(2 \times 7 + 1) = 75

cm²