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ALGEBRAIC - INEQUALITIES

All Levels

Graph each of the following inequalities on the number line given.

𝑥 < 4, where 𝑥 ∈ ℕ.

2016 JCHL Paper 1 – Question 7 (b) (i)

𝑥 ∈ ℕℕ are the natural numbers. They are whole numbers and so are represented with full dots on each number.

𝑥 ∈ ℤℤ are the integers. They are positive and negative whole numbers and are represented by full dots.

𝑥 ∈ ℝℝ are the real numbers and include all the fractions and decimals. These are represented by a shaded number line.


𝑥 < 4, where 𝑥 ∈ ℤ.

2016 JCHL Paper 1 – Question 7 (b) (ii)





𝑥 < 4, where 𝑥 ∈ ℝ.

2016 JCHL Paper 1 – Question 7 (b) (iii)




5 marks in total for (i), (ii) AND (iii)!

Complete the inequality in 𝑛 below so that it has the solution set shown.

2015 JCHL Paper 1 – Question 8 (a)

Complete the inequality in 𝑥 below so that there is only one possible value of 𝑥,where 𝑥 ∈ ℝ.

(b)

2 4

10.1 10.1



15 marks

Note: There are multiple answers for both of these questions (there is actually an infinite amount of solutions!)

Examples

(a)2 ≤ 𝑛 ≤ 4.71.5 ≤ 𝑛 ≤ 3.5

(b)2 ≤ 𝑥 ≤ 217.3 ≤ 𝑥 ≤ 17.3

≤ 𝑥 ≤€25 €50

≤ 𝑦 ≤€35 €60

Niamh is in a clothes shop and has a voucher which she must use.The voucher gives a €10 reduction when buying goods to the value of at least €35.She also has €50 cash.Write down an inequality in 𝑥 to show the range of cash that she could spend in the shop.

2015 Sample JCHL Paper 1 – Question 6 (a)

Niamh buys one item of clothing in the shop, using the voucher as she does so. Write an inequality in 𝑦 to show the range of possible prices that this item could have been, before the €10 reduction is applied.

(b)

−17 ≤ 1 − 3𝑥 < 13

Solve the following inequality and show the solution on the number line.−17 ≤ 1 − 3𝑥 < 13, 𝑥 ∈ ℤ

2014 JCHL Paper 1 – Question 5

1 − 3𝑥 < 131 − 13 < 3𝑥−12 < 3𝑥−4 < 𝑥

−17 ≤ 1 − 3𝑥3𝑥 ≤ 1 + 173𝑥 ≤ 18𝑥 ≤ 6


Compound Inequalities (where there is 2 inequality signs) can be split into two sums which we solve separately.

We treat inequalities like equations but need to be a little more careful when dividing by negative numbers.

Try and collect the 𝑥 terms to the side where they will be positive.

If you end up with a negative 𝑥 term and divide by that negative you must change the direction of the inequality.

−4 < 𝑥 ≤ 6

10 marks

Solve the inequality −2 < 5𝑥 + 3 ≤ 18, 𝑥 ∈ ℝ.

2014 Sample JCHL Paper 1 – Question 12 (a) (i)

Graph your solution on the number line below.

(ii)

−2 < 5𝑥 + 3 ≤ 18

5𝑥 + 3 ≤ 185𝑥 ≤ 18 − 35𝑥 ≤ 15𝑥 ≤ 3

−2 < 5𝑥 + 3−2 − 3 < 5𝑥−5 < 5𝑥−1 < 𝑥


−1 < 𝑥 ≤ 3

Hollow circle as it is NOT equal to −𝟏 and a full filled circle as it IS equal to 3.

2 ≤ 𝑥 < 8

2, 3, 4, 5, 6, 7

Solve the following inequality and show the solution on the number line.

−2 ≤1

2𝑥 − 3 < 1, 𝑥 ∈ ℕ.


−2 ≤1

2𝑥 − 3 < 1

1

2𝑥 − 3 < 1

1

2𝑥 < 1 + 3

1

2𝑥 < 4

𝑥 < 8

−2 ≤1

2𝑥 − 3

−2 + 3 ≤1

2𝑥

1 ≤1

2𝑥

2 ≤ 𝑥

𝑥 ∈ Nℕ are the natural numbers. They are whole numbers and so are represented with full dots on each number.

5 marks

32𝑥 ≤ 3000 − 800

32𝑥 ≤ 3000 − 800

32𝑥 ≤ 2200

𝑥 ≤2200

32

𝑥 ≤ €68.75

Josephine hopes to go to college. She has saved €3000. She will attend college for 32 weeks in her first year. She plans to have at least €800 left at the end of the year.If she spends €𝑥 each week, write an inequality to represent her spending during the year.


Hence, or otherwise, find the maximum amount Josephine can spend each week.


The amount that she spends in the 32 weeks, 𝟑𝟐𝒙, must be less than or equal to the amount she saved minus the amount she wants to have left.

10 marks5 marks

2 − 3𝑥 ≥ −62 + 6 ≥ 3𝑥8 ≥ 3𝑥8

3≥ 𝑥

List all the values of 𝑥 that satisfy the inequality 2 − 3𝑥 ≥ −6, 𝑥 ∈ 𝑁.

2015 LCOL Paper 1 – Question 3 (b)

Treat as if it were an equation collecting ‘like’ terms on the same side.

The solution is the set of Natural Numbers

(positive whole numbers) less than 𝟐𝟐

𝟑

𝑥 ∈ 1,2

We treat inequalities like equations but need to be a little more careful when dividing by negative numbers.

Try and collect the 𝑥 terms to the side where they will be positive.

If you end up with a negative 𝑥 term and divide by that negative you must change the direction of the inequality.

5 marks

Solve the following inequality, and show the solution set on the number line below.

5 −3

4𝑥 ≤

19

8

2011 LCOL Paper 1 – Question 4 (b)

5 −3

4𝑥 ≤

19

8

5 −19

8≤

3

4𝑥

21

8≤

3

4𝑥

21834

≤ 𝑥

3.5 ≤ 𝑥

For the number line we must ASSUME that 𝒙 ∈ ℝ as we are not told otherwise.

10 marks

OLD COURSE

All Levels

Find the largest possible value of 𝑛 such that 5𝑛 + 48 > 8𝑛 − 6, 𝑛 ∈ ℕ.


5𝑛 + 48 > 8𝑛 − 6

48 + 6 > 8𝑛 − 5𝑛

54 > 3𝑛

18 > 𝑛

𝒏 is less than 18 so the largest number 𝒏 could be is 𝒏 = 𝟏𝟕

4 − 𝑥 ≥ 2𝑥 − 5

4 + 5 ≥ 2𝑥 + 𝑥

9 ≥ 3𝑥

3 ≥ 𝑥

0 1 2 3 4 5 6

Graph on the number line the solution set of4 − 𝑥 ≥ 2𝑥 − 5, 𝑥 ∈ ℕ.



−2𝑥 + 1 > −7

1 + 7 > 2𝑥

8 > 2𝑥

4 > 𝑥

Graph on the number line the solution set of −2𝑥 + 1 > −7, 𝑥 ∈ ℕ.



List the elements of the solution set of−5 ≤ 3𝑥 − 2 < 7, 𝑥 ∈ 𝒁.


−5 ≤ 3𝑥 − 2 < 7

−5 ≤ 3𝑥 − 2−5 + 2 ≤ 3𝑥−3 ≤ 3𝑥−1 ≤ 𝑥

3𝑥 − 2 < 73𝑥 < 7 + 23𝑥 < 9𝑥 < 3

Solution set is all the integers greater or equal to −𝟏 and less than 3

−1, 0, 1, 2

Graph on the number line the solution set of−3 < 4𝑥 + 7 ≤ 23, 𝑥 ∈ 𝑹.


−3 < 4𝑥 + 7 ≤ 23

−3 < 4𝑥 + 7−3 − 7 < 4𝑥−10 < 4𝑥−2.5 < 𝑥

4𝑥 + 7 ≤ 234𝑥 ≤ 23 − 74𝑥 ≤ 16𝑥 ≤ 4

−2.5 < 𝑥 ≤ 4

Hollow circle as it is NOT equal to −𝟐. 𝟓 and a full filled circle as it IS equal to 𝟒.


Graph on the number line the solution set of −98 ≤ 10 − 12𝑥, 𝑥 ∈ 𝑵.


−98 ≤ 10 − 12𝑥

12𝑥 ≤ 10 + 98

12𝑥 ≤ 108

𝑥 ≤ 9


Find A, the solution set of 3𝑥 − 5 < 7, 𝑥 ∈ ℤ.

2011 LCOL Paper 1 – Question 2 (b) (i)

Find B, the solution set of −2−3𝑥

4≤ 1, 𝑥 ∈ ℤ.

2011 LCOL Paper 1 – Question 2 (b) (ii)

List the elements of 𝐴 ∩ 𝐵.

2011 LCOL Paper 1 – Question 2 (b) (iii)

3𝑥 − 5 < 73𝑥 < 7 + 53𝑥 < 12𝑥 < 4

−2 − 3𝑥

4≤ 1

−2 − 3𝑥 ≤ 4−2 − 4 ≤ 3𝑥−6 ≤ 3𝑥−2 ≤ 𝑥

−2 ≤ 𝑥 < 4

−2, −1, 0, 1, 2, 3

Find the value which satisfy2 3 + 4𝑥 ≤ 22, where 𝑥 ∈ ℕ.

2010 LCOL Paper 1 – Question 2 (a)

2 3 + 4𝑥 ≤ 22

6 + 8𝑥 ≤ 22

8𝑥 ≤ 22 − 6

8𝑥 ≤ 16

𝑥 ≤ 2

𝑥 = 1, 2

Find the solution set of 4𝑥 − 15 < 1, 𝑥 ∈ 𝐍.


4𝑥 − 15 < 1

4𝑥 < 1 + 15

4𝑥 < 16

𝑥 < 4

𝑥 = 1, 2, 3

Find 𝐴, the solution set of 3𝑥 − 2 ≤ 4, 𝑥 ∈ 𝐙.

2005 LCOL Paper 1 – Question 3 (b) (i)

Find 𝐵, the solution set of 1 − 3𝑥

2< 5, 𝑥 ∈ 𝐙.

2005 LCOL Paper 1 – Question 3 (b) (ii)

List the elements of 𝐴 ∩ 𝐵.

2005 LCOL Paper 1 – Question 3 (b) (iii)

3𝑥 − 2 ≤ 43𝑥 ≤ 4 + 23𝑥 ≤ 6𝑥 ≤ 2

1 − 3𝑥

2< 5

1 − 3𝑥 < 101 − 10 < 3𝑥−9 < 3𝑥−3 < 𝑥

−3 < 𝑥 ≤ 2

−2, −1, 0, 1, 2

Find the solution set of5𝑥 − 3 < 12 , 𝑥 ∈ 𝐍.


5𝑥 − 3 < 12

5𝑥 < 12 + 3

5𝑥 < 15

𝑥 < 3

𝑥 = 1, 2

Solve the inequality 5𝑥 + 1 ≥ 4𝑥 − 3 for 𝑥 ∈ 𝑅 and illustrate the solution set on a number line.


5𝑥 + 1 ≥ 4𝑥 − 3

5𝑥 − 4𝑥 ≥ −3 − 1

𝑥 ≥ −4


Find the solution set of 11 − 2𝑛 > 3, 𝑛 ∈ N.


11 − 2𝑛 > 311 − 3 > 2𝑛8 > 2𝑛4 > 𝑛

𝑛 = 1, 2, 3

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