**5. ****Algebra 2**

**5.1 – Algebraic Fractions**

- Simplifying Fractions:

- Can simplify algebraic fractions using division
- Where possible factorise the numerator and denominator, then cancel common factors.
- E.g. 5x
^{2}– 245

- Addition and Subtraction of Algebraic Fractions:

- To add or subtract algebraic fractions:

- Find the lowest common multiple of the denominators
- Express fractions in terms of the L.C.M
- Simplify numerators
- Add numerators

- E.g. 5x + 2x
**(Lowest common denominator of 6 and 9 is 18)**

**5.2 – Changing the Subject of a Formula**

- When changing the subject of a formula, you rearrange it so that you have a different subject
- To do this, move a term from one side of the equal sign to the other side and change the operation to do the opposite (inverse operation).

- E.g. Make ‘y’ the subject of the formula x(y – z) = a

x(y – z) = a

xy – xz = a

xy = a + xz

y = a + xz

x

**5.3 – Variation**

- Direct Variation:

- Several ways of expressing a relationship between two quantities ‘x’ and ‘y’
- E.g.

- ‘x’ varies directly as ‘y’
- ‘x’ varies as ‘y’
- ‘x’ is proportional to ‘y’

- These all mean the same and can be written in symbols:
- x ∝ y

- ‘∝’ sign can be replaced with ‘= k’ where k is a constant:
- x = ky

- E.g. Suppose x = 6 when y = 24

6 = k × 24

k = 6/24

k = ¼

You can then write x = ¼ × y, this enables you to find any value of ‘x’ for any value of ‘y’ and vice versa.

- Inverse Variation:

- Several ways of expressing an inverse relationship between two variable ‘x’ and ‘y’
- E.g.

- ‘x’ is inversely proportional to ‘y’
- ‘x’ varies inversely as ‘y’

- These all mean the same and can be written in symbols:
- x ∝ 1/y

- ‘∝’ sign can be replaced with ‘= k’ where k is a constant:
- x = k × 1/y

**5.4 – Indices**

- Rules of Indices:

- a
^{n }× a^{m }= a^{n + m}

- E.g. 4
^{2 }× 4^{4 }= 4^{6}

- a
^{n }÷ a^{m}= a^{n – m }

- E.g. 3
^{5}÷ 3^{2 }= 3^{3}

- (a
^{n})^{m}= a^{n m}

- E.g. (2
^{4})^{5 }= 2^{20}

- a
^{-n }= 1/a^{n}^{ }

- E.g. 6
^{-2}= 1/6^{2}

- a
^{1}^{/n }means the nth root of a

- E.g. 8
^{1/2 }= √8

- a
^{m/n }means the nth root of a raised to the power m

- E.g. 5
^{3/2}= (√5 )^{3}

**5.5 – Inequalities**

**<**(less than)

- E.g. x < 5 (x is less than 5)

**>**(greater than)

- E.g. x > 8 (x is greater than 8)

**≤**(less than or equal to)

- E.g. x ≤ 12 (x is less than or equal to 12)

**≥**(greater than or equal to)

- E.g. x ≥ 4 (x is greater than or equal to 4)

- Solving Inequalities:

- You use the same procedure used for solving equations.
- Except when you multiply or divide by a negative number the inequality will be reversed

- E.g. 12 – 3x < 27

-3x < 15 (Subtract 12 from both sides)

x > -5 (Divide both sides by -3)

- The Number Line:

- You may need to represent inequalities on a number line.

- is used for < and > and means the end value is
*not*included - is used for ≤ and ≥ and means the end value is included

**5.6 – Linear Programming**

- In most linear programming problems, there will be two stages:

- Interpret the information given as a series of simultaneous inequalities and display them graphically.

Investigate some characteristic of the points in the unshaded solution set.