GCSE Algebra Higher Notebook » On-line test

Year 10/11 maths revision

Inequalities  Menu

These are equations using greater than > or less than <. In each case, the sign opens towards the larger number.
y ≤ 6 means "y is less than or equal to 6"
z ≥0 means "z is greater than or equal to 0".


Write down the integer values of x that satisfy the inequality:  – 4 ≤ x < 2 

x can be –4, –3, –2, –1, 0 or 1


Show inequalities on a number line:
2x + 1 ≥ 7 → 2x ≥ 6 or x ≥3
inequality

1 < 3x - 5 ≤ 10 → 6 < 3x ≤ 15 → 2 < x ≤5
inequality

Solving an inequality
Change the inequality sign into an equals sign and treat the inequality like an equations

e.g. 5a – 16 < 8 – a   →  5a – 16 = 8 – a
  → 6a – 16 = 8  →   6a = 24
so putting back <  →  a < 4

Remember: if you change the sign the inequality sign changes.
e.g. If –5x < 30, then x > –6.


Q: Find integer values for x:    – 2 < x ≤ 5 

Q: Show y on a number line :   1 > y ≥ – 5 

Q: Solve inequality for a:  6a + 6 > 48 – a  

ANS

Direct, Inversely Proportional  Menu
proportional

When a quantity y varies directly with a quantity x, it means when x changes, y changes at the same rate.

We write this as y ∝ x, then as y = k × x.
We say y is directly proportional to x and k is a constant.

The graph shows this and the gradient is k.
If y = 2x the values are directly proportional and the constant or gradient is 2.



inversely

When a quantity y varies inversely with a quantity x, it means when y goes up, x goes down at the same rate.

We write this as reciprocal
If y = 2/x the values are inversly proportional and the constant is 2.


In proportional questions you need to find the constant k.

Q1. For a given spring, F is directly proportional to its length L.
If F = 32 when the spring is stretched 8 cm, find the constant k and a formulae for F in terms of L.

Q2. What is the value of F when the spring has stretched 11 cm?

Q3. 4 people can build a wall in 3 hours.How long will it take 6 people to build it?

ANS

Simultaneous Equations  Menu

Look at the pencils and rubbers below. Can you work out the cost for each pencil and rubber?

simultaneous

Writing this in algebra we get:
R + 3P = 40
R + 2P = 30
Can you see that one pencil costs 10p?
and the cost of one rubber is 10p.
You have solved some simultaneous equations by 'eliminating' the rubbers.


How to eliminate one unknown from:
 x + y = 10
 x – y = 2

If we add the two equations together, y will disappear leaving 2x = 12 so x =6.
Now we put x = 6 into one of the equations
6 + y =10 so y = 4

How to eliminate one unknown from:
 3x + y = 9
   x + y = 5

If we subtract the two equations, y will disappear leaving 2x = 4 so x = 2.
Now we put x = 2 into one of the equations
6 + y = 9 so y = 3

How to eliminate one unknown from:
 3x + 2y = 12
   x + y = 5

This time start by multiplying the bottom line by 2 to give:  2x + 2y = 10
Subtract the two equations
 3x + 2y = 12
 2x + 2y = 10

and 2y disappears leaving x = 2 .
Put x = 2 into one of the equations
6 + 2y = 12 so y = 3

The trick is to get the x or the y terms to be the same in both equations.

Q1. Solve:
 3x + 2y = 19
    x + y = 7   

Q2. Solve:
 3x – 2y = 22
 3x + 2y = 14   

ANS

Quadratic Equations  Menu

The name Quadratic comes from "quad" or square, because of the x2 term.

The usual form of a quadratic equation is ax2 + bx + c = 0 where a, b and c are numbers.

Q: Which of these are quadratic equations:

  • x2 + 2x = 8
  • x(x – 1) = 4
  • 2x2 + 9 = 0  ANS

Finding Solutions

The "solutions" (usually 2) to a quadratic equation are when it equals zero and are called "roots".
These are the values of x that when substituted in the equation make zero.

There are three ways to solve quadratics:
  • Factorising
  • Completing the Square
  • Using a Formulae

Factorising
x2 + 2x – 8 can be factored as (x + 4)(x – 2)
The rule is that you need two numbers that multiply to make the number at the end and that can also be added or subtracted to make the number before the x.

If x2 + 2x – 8 = 0 then (x + 4)(x – 2) = 0
so x + 4 = 0 or x – 2 = 0 and x = – 4 or +2

x2 – 10x + 25 is factored as (x – 5)(x – 5)
This is called a perfect square.

Try factorising these:

 x2 – 3x + 2 = 0        x2 – x – 30 = 0 
 x2 + 12x + 36 = 0    x2 – 2x + 1 = 0 

ANS

Complete the square  Menu

Complete the square for x2 + 10x + 24

We want to make x2 + 10x + 24 look like:
(x + a)2 – b, where a and b are numbers.


Make x2 + 10x → (x + 5)2 by halving the 10
(x + 5)2 → (x + 5)2 – 25  by removing the 52


So x2 + 10x = (x + 5)2 – 25 (check it)
and finally put back the 24


So x2 + 10x + 24 = (x + 5)2 – 1


To solve x2 + 10x + 24 = 0
we solve (x + 5)2 – 1 = 0


Rearrange  (x + 5)2 = 1
square root  x + 5 = √1
rearrange so x = ± 1 – 5
x = – 4 or – 6

Complete the square and solve:

 y2 – 12y + 5 = 0 give answer as a √ surd

ANS

Quadratic Formula  Menu

The standard form of a quadratic equation is: ax2 + bx + c = 0

Use the formula below to solve it:

– b ± √(b² – 4ac) / 2a

The ± means there may be two solutions.
b2 – 4ac is called the discriminant (AS level)


For 3x2 + 5x – 8 = 0   a = 3, b = 5, c = –8
So x is :

– 5 ± √(5² – 4 ×3 ×–8) / 2 × 3
– 5 ± √(25 +96) / 6
– 5 ± √(121) / 6
– 5 ± 11 / 6
x = – 2.66 or 1

Q: Using the quadratic formula solve:

 5y2 + 6y + 1 = 0     ANS

Equation of a straight line Menu

Straight line graph equation is y = mx + c

The m is the gradient of the line (how steep it is) and the c is where the line crosses the y axis

In this graph c = –1,

... but how do we work out the gradient?


Finding the gradient (steepness) gradient


Draw a horizontal and vertical line and with the graph line as the longest side make a right angled triangle.


The gradient is the y-distance ÷ x-distance and is ⅔

The gradient of the line graph is 1 ÷ 1 = 1
So, the equation of the line is y = x –1


Q: What is the equation of this line?

graph ANS

Sequences Menu

A Sequence is a list of things (usually numbers, called terms) in some order. We call the position of each term the nth position. nth means, 1st, 2nd, 3rd position etc

sequence

This sequence goes up by two each time.

So the rule is 'start at three and increase by two'

We can change the rule into a formula (nth term formula) for the sequence above.

Method

Find the jump - in this case it is 2.

Write it as 2n

Then replace n with 1, 2, 3 etc to show the term for each position in the sequence.


Term position: 1st   2nd  3rd  4th
So n  =  1    2     3     4
and 2n = 2    4     6     8
Our sequence = 3    5     7     9

If we compare our sequence with the 2n sequence we can see that we have added 1

∴ nth term = 2n + 1

Q1: What is the nth term: 6, 11, 16, 21 ..?

Q2: What is the nth term : 4, 11, 18, 25 ..?

Q3: What is the 100th term : 3, 7, 11, 15..?

ANS

Algebraic Fractions Menu
algebraic fraction 1

An algebraic fraction is just like a normal fraction but with a letter included.

This example is to add half x and half y.

algebraic fraction 2

You can use the smile and kiss method of adding normal fractions.
The smile multiplies the bottom two values and the cross multiplies across from bottom right to top left and bottom left to top right.


So the answer is: algebraic fraction 3

Smile and Kiss  :algebraic fraction 4

Two more examples which show the process.
two examples


Q: Simplify more algebraic fractions ANS

Surds Menu

A rational number is a number with a pattern:
e.g. 1/3 ;  √16;  2.12121212

A surd is an irrational number with no pattern.
e.g. √5 = 2.236067977...

Q: Which of these is rational:
4/9     √5    0.6    √36  ANS

A surd has an infinite number of non-recurring decimals.

A surd can be rationalise by multiplying it by itself: √5 × √5 = 5

Q: Simplify
 ,    ,  
ANS

Powers (exponents) Menu

We started powers or indices in basics

Fractional powers

A power of ½ is a square root

4 ½ = √4 = ± 2    100 ½ = √100 = ± 10

A power of ⅓ is cube root

8 = ∛8 = 2    27 = ∛27 = 3

We add powers when we multiply

 x½ × x½ = x(½ + ½) = x1= x
 x½ × x¼ = x(½ + ¼) = x¾

So to rationalise surds:

√5 × √5 = 5 ½ × 5 ½ = 5

We subtract powers when we divide

 x¾ ÷ x½ = x(¾ – ½)  =   x¼
 x½ ÷ x¼ = x(½ – ¼) = x¼

Negative or reciprocal powers

negative powers 2–1 = 1 ÷ 21 = 0.5
4 –½ = 1 ÷ 4½ = 1 ÷ 2 = 0.5
8 –⅓ = 1 ÷ 8 = 1 ÷ 2 = 0.5

Use the same power rules

 x–½ ×  x½ = x(– ½ + ½) = x0 = 1
 x–½ ÷  x½ = x(– ½ – ½) = x–1

Q1 Simplify: 8¾ ÷ 8½ × 8½ × 8¼

Q2: Simplify: 4–⅛ × 4–⅜ ×  4   ANS

Using Algebra Menu
4 sided shape

The angles in this 4 sided shape are given in terms of b.
How can you find b? What do you know?

All the angles add up to 360° so:
b–45° + b+20° + b+30° + 70° = 360°
3b + 75° = 360° so 3b = 285° and b = 95°

Now look at this shape

triangle

Write an equation for the triangle

Solve it to work out x    ANS