IGCSE Further Maths/C1 Inequalities

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 4th September 2015

Objectives: Be able to solve both linear and quadratic inequalities. Be able to manipulate inequalities (including squared terms).

RECAP :: Linear Inequalities

Solve Jan 2013 Paper 2

𝑑>5

Work out the greatest integer value of that satisfies the inequality

Thus greatest integer is 2.

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Bromonology: Remember that ‘linear’ just means that if we plotted the equation/inequality we’d end up with a straight line.

June 2012 Paper 1

Solve

𝟔<𝟑 𝒙 ≤𝟏𝟖?

Manipulating Inequalities

−3 ≤𝑥≤2

What is the smallest value of ?

What is the largest value of ?

Hence determine an inequality for .

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Test Your Understanding

−1≤ 𝑥≤4→0 ≤𝑥2≤16 and (a) Work out an inequality for .

(b) Work out an inequality for

June 2012 Paper 1

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Further Example

Given that and , work out an inequality for .

What’s the least can be?

What’s the greatest can be?

Thus inequality for :

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Exercise 1Solve the following:

Given that and , work out the inequality for .

Given that and work out inequalities for:

Given and , decide whether the following statements are ALWAYS TRUE< SOMETIMES TRUE, or NEVER TRUE. Always true Never true Sometimes true. Never true. Sometimes true.

[June 2013 Paper 2] is an integer such that . is an integer such that .(a) What is the highest possible value of

16(b) What is the lowest possible value of . -1

Given that state a value of for which:e.g. 1.5e.g. 0.5e.g. -0.5

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2

3

4

5

6

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abc

d

e

f

ghi

abcde

abc

Quadratic Inequalities

Solve

! Step 1: Get 0 on one side.

𝑥2−4 𝑥−5<0 ! Step 2: Factorise.

! Step 3: Sketch .

𝑥

𝑦

-1 5

! Step 4: Identify parts of line where value (i.e. LHS of inequality) satisfies inequality.

𝑦=(𝑥+1 ) (𝑥−5 )

Since , we’re interested in the parts of the line where .

Therefore: ?Step 4?

Step 2 ?

Step 3 ?

Quadratic Inequalities

Solve

! Step 1: Get 0 on one side.

𝑥2−4 𝑥−5>0 ! Step 2: Factorise.

! Step 3: Sketch .

𝑥

𝑦

-1 5

! Step 4: Identify parts of line where value (i.e. LHS of inequality) satisfies inequality.

𝑦=(𝑥+1 ) (𝑥−5 )

Since , we’re interested in the parts of the line where .

Therefore: orStep 4?

Now suppose we changed < for >…

Test Your Understanding

Solve Solve

(𝑥+3 ) (𝑥−2 )≤0

𝑥

𝑦

-3 2

(note that has to be consistent with original question)

𝑥2+2𝑥−3>0

𝑥

𝑦

-3 1

or

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Exercise 2Solve the following inequalities:(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

The area of the square is less than the area of the rectangle. Work out an inequality for .

𝑥+1

𝑥+1

2 𝑥−1

𝑥−1(but clearly can’t be less than 0)

1

2

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C1 Discriminants

We now (hopefully!) have the sufficient skills to tackle more questions concerning discriminants:

Edexcel C1 Jan 2013

Discriminant:

After sketching:

Reminder:No solutions: Equal solutions: Distinct solutions:

a

b

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Test Your Understanding

Edexcel C1 Jan 2011

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Combining Inequalities

Edexcel C1 June 2009

a) b)

c) It may help to draw number lines for both and combine. Otherwise use common sense!

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Exercise 3

Edexcel C1 Jan 2010 Q10

Edexcel C1 Jan 2009 Q7

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