Equations and Inequalities

Introduction

This Chapter focuses on solving Equations and Inequalities

It will also make use of the work we have done so far on Quadratic Functions and graphs

Equations and Inequalities

Simultaneous Equations

You need to be able to solve Simultaneous Equations by Elimination.

Remember that the 2 equations can also be drawn on a graph, and the solutions are where they cross.

However, this method is less accurate when we start needing decimal/fractional answers.

GENERAL RULE

If what you’re cancelling have different signs, add. If they have the SAME sign, SUBTRACT!

Example

Solve the following Simultaneous Equations by Elimination

2 3 8x y 3 23x y

1

2

2 3 8x y 9 3 69x y x3

Add

11 77x

7x

3 23x y

Substitute x in to ‘2’

21 23y

2y

2

Equations and InequalitiesSimultaneous Equations

You need to be able to solve Simultaneous Equations by Elimination.

Remember that the 2 equations can also be drawn on a graph, and the solutions are where they cross.

However, this method is less accurate when we start needing decimal/fractional answers.

GENERAL RULE

If what you’re cancelling have different signs, add. If they have the SAME sign, SUBTRACT!

ExampleSolve the following Simultaneous

Equations by Elimination

4 5 4x y 6 2 25x y

1

2

12 15 12x y 12 4 50x y x2

Subtract

19 38y 2y

6 2 25x y

Substitute y in to ‘2’

3.5x

2

x3

6 4 25x


You need to be able to solve Simultaneous Equations by Substitution.

This involves using one equation to write y ‘in terms of x’ or vice versa. This is then substituted into the other equation.


Equations by Substitution

2 1x y 4 2 30x y

1

2

2 1y x Rearrang

e

Replace the ‘y’ in equation 2, with ‘2x – 1’

(22 01 3)4x x

2 4 2 30yx

4 4 2 30x x

8 2 30x

8 28x

3.5x

8y

Replace y

Expand

Sub into 1 or 2


You will need the substitution method when one of the equations is quadratic.

You will end up with 0, 1 or 2 answers as with any Quadratic.

This means you will either get 0, 1 or 2 pairs of answers



2 3x y 2 3 10x xy

1

2

3 2x y Re-arrange

Replace the ‘x’ in equation 2, with ‘3 – 2y’

2 3 10x xy 223 2 3 2( ) 3 ( ) 10yy y

2 29 12 4 9 6 10y y y y 22 3 1 0y y 22 3 1 0y y

(2 1)( 1) 0y y 1

or 12

y y

Expand Brackets

Simplify

Multiply by -1

Factorise

Solve

2 3x y

Sub each value for y into one of the equations

( 1) 3x ( 2) 3x 4x 5x

y = -1/2y = -1

x = 4, y = -1/2

x = 5, y = -1







3 2 1x y 2 2 25x y

1

2

3 1

2

xy

Re-arrange

Replace the ‘y’ in equation 2, with ‘3x – 1/2’

2 2 25x y 2

22 3 1

225

xx

22 9 6 1

254

x xx

2 24 9 6 1 100x x x 213 6 99 0x x

Replace y

Square top and bottom separately

Multiply each part by 4

Group on one side

213 6 99 0x x

(13 33)( 3) 0x x

33

13x 3x or

Factorise

Solve







3 2 1x y 2 2 25x y

1

2

3 1

2

xy

Re-arrange

33

13x 3x or

3 1

2

xy

333 1

132

y

991

132

y

112132

y

112

26y

56

13y

3 3 1

2y

9 1

2y

8

2y

4y

x = -33/13

x = 3

x = -33/13, y = -56/13x = 3, y = 4


Solving Inequalities

You need to be able to solve Linear Inequalities, sometimes more than one together.

An Inequality will give a range of possible answers, rather than specific values (like an Equation would).

You can solve them in the same way as a Linear Equation.

Only difference: When you multiply or divide by a negative, you must reverse the sign

> <

ExampleFind the set of values of x for

which:

2 5 7x

2 5 7x

2 12x

6x

Add 5

Divide by 2







> <


which:

5 9 20x x

Subtract x

Subtract 9

5 9 20x x

4 9 20x

4 11x

2.75xDivide by

4







> <


which:

12 3 27x

Subtract 12

Divide by 3

Multiply by -1

REVERSES THE SIGN

12 3 27x

3 15x

5x

5x







> <


which:

3( 5) 5 2( 8)x x

Expand brackets (careful with negatives)

Add 2x and group

Add 15

3 15 5 2 16x x

3( 5) 5 2( 8)x x

5 15 21x

5 36x

7.2x Divide by 5







> <


which:

3 5 8x x

Subtract x

5 8x x and

3 5 8x x

2 5 8x

2 13x

6.5x

5 8x x

4 8x

2x Add 5

Divide by 2

Subtract x

Divide by 4

-4 -2 6420 8 10

2 6.5x

x < 6.5x > -2







> <


which:

5 1x x

Add x

15 3 5 2x x and

5 1x x 15 3 5 2x x

Add 5

Divide by 2

Add 3x

Minus 5

2 5 1x

2 6x

3x

15 5 5x

10 5x

2 xDivide by 5

-4 -2 6420 8 10 x > 3

x < 2

No answers that work for both…







> <


which:

4 7 3x

Subtract 7

17 11 2x and

4 7 3x 17 11 2x

Divide by 4

Subtract 11

Divide by 2

4 4x

1x

6 2x

3 x

-4 -2 6420 8 10 x > -1

x > 3

3x

Equations and InequalitiesQuadratic Inequalities

To solve a Quadratic Inequality, you need to:

1) Solve the Quadratic Equation

2) Sketch a graph of the Equation

3) Decide which is the required set of values

Remember that the solutions are where the graph crosses the x-axis

The graph will be u-shaped. Where it crosses the y-axis does not matter for this topic

Then think about which area satisfies the original inequality


which:2 4 5 0x x

2 4 5 0x x

( 1)( 5) 0x x Factorise

1 or 5x x y

x-1 5

We want values below 0

1 5x







The graph will be u-shaped. Where it crosses the y-axis does not matter for this topic



which:2 4 5 0x x

2 4 5 0x x

( 1)( 5) 0x x Factorise

1 or 5x x y

x-1 5

We want values above 0

1 or 5x x

Separate sections

mean separate

inequalities







The graph will be n-shaped, looking at the original equation



which:23 5 2 0x x

23 5 2 0x x 22 5 3 0x x

Multiply by -1

0.5 or 3x x y

x-3 0.

5

We want values below 0

(2 1)( 3) 0x x Factorise

3 or 0.5x x

Summary

We have looked at solving Simultaneous Equations, including Quadratics

We have seen how to solve Inequalities

We have seen how to use graphs to solve Quadratic Inequalities

Chapter 4 (sketching curves)

Equations and Inequalities

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