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
Equations and InequalitiesSimultaneous Equations
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.
ExampleSolve the following Simultaneous
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
Equations and InequalitiesSimultaneous Equations
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
ExampleSolve the following Simultaneous
Equations by Substitution
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
Equations and InequalitiesSimultaneous Equations
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
ExampleSolve the following Simultaneous
Equations by Substitution
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
Equations and InequalitiesSimultaneous Equations
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
ExampleSolve the following Simultaneous
Equations by Substitution
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
Equations and Inequalities
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
Equations and Inequalities
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:
5 9 20x x
Subtract x
Subtract 9
5 9 20x x
4 9 20x
4 11x
2.75xDivide by
4
Equations and Inequalities
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:
12 3 27x
Subtract 12
Divide by 3
Multiply by -1
REVERSES THE SIGN
12 3 27x
3 15x
5x
5x
Equations and Inequalities
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:
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
Equations and Inequalities
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:
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
Equations and Inequalities
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:
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…
Equations and Inequalities
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:
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
ExampleFind the set of values of x for
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
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
ExampleFind the set of values of x for
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
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 n-shaped, looking at the original equation
Then think about which area satisfies the original inequality
ExampleFind the set of values of x for
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)