Aims:• To practice sketching graphs of rational functions• To be able to solve inequalities by sketching and observation.• To be able to solve inequalities by using an algebraic method

Graphs Lesson 4

Starter

Example 2 Solve (x - 2)(3x – 1) ≤ 0

Solve x2 + x – 3 > 4x + 1.

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Remember: When both sides of an inequality are multiplied or divided by a negative number the inequality is reversed.

–3 < 5

Multiply both sides by –1:

3 < –5

So we have to reverse the inequality sign:

3 > –5

Rules of Inequalities

This means that if we were trying to solve the inequality

We could not just multiply both sides by as we donot know if this is negative. What we do have to do, is multiply it by , which has to be positive.

3dcxbax

2dcx dcx

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Methods to Solve Inequalities

There are two methods we will look at to solve rational function inequalities.

Method 1Algebraic method: multiply both sides by the denominator squared.Then solve the quadratic inequality.

Method 2Sketching method: sketch the rational function and the line y = 2, then look to see where the curve is greater than the line y = 2. Points of intersection will need to be found.

2123

xx

212 x

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Example 1

Solve the inequality 2123

xx

Method 1

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Method 2

1. Find the intercepts with the axes

2. Find the vertical asymptotes

3. Examine the behaviour as x tends to

Example 1

Solve the inequality 2123

xx

123

xxy

x y

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4. Draw line y = 2 see where they cross

2123 Solve

xx

2123 So

xx when

Example 2

Solve the inequality 2)2)(1()4)(1(

xxxx

Method 1

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Example 2

Solve the inequality 2)2)(1()4)(1(

xxxx

Method 2

1. Find the intercepts with the axes

2. Find the vertical asymptotes

3. Examine the behaviour as x tends to

)2)(1()4)(1(

xxxxy

First sketch

x y

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4. Draw line y = 2 see where they cross

when2)2)(1()4)(1( So

xxxx

2)2)(1()4)(1( Solve

xxxx

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1. Solve the inequality (you chose your favourite method or show me both?)

On w/b

223

xx

Do ex 5G page 71. then revision ex 5

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