## 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.     5x2 – 245
• Addition and Subtraction of Algebraic Fractions:
• To add or subtract algebraic fractions:
1. Find the lowest common multiple of the denominators
2. Express fractions in terms of the L.C.M
3. Simplify 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:
1. an × am = an + m
• E.g. 42 × 44 = 46
1. an ÷ am = an – m
• E.g. 35 ÷ 32 = 33
1. (an)m = an m
• E.g. (24)5 = 220
1. a-n = 1/an
• E.g. 6-2 = 1/62
1. a1/n  means the nth root of a
• E.g. 81/2 = √8
1. am/n  means the nth root of a raised to the power m
• E.g. 53/2 = (√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:
1. 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.

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