Year 9 Maths Chapter 2 - Expressions, equations and inequalities

Complete

You will want to refer to Year 7 Maths Chapter 9 and Year 8 Maths Chapter 2 for basic equations. This section is more advanced, and omits basic concepts.

More on equations are located in Chapter 8

Table of Contents

• Inequalities
  • EXAMPLE
  • Inequalities on a number line
• Simulataneous Equations
  • Substitution
    • Example
  • Elimination
    • Example
    • Example
• ax² = c
  • Example
• Point of Intersection

Inequalities

• Similar to an equation
• Always give answer in terms of the pronumeral.
• When dividing by a negative, reverse the inequality sign

EXAMPLE

4 - 5x < 1
-5 x < -3
5x > 3 # we divide by -1, and reverse the inequality sign accordingly
x > ⅗

Inequalities on a number line

• Open circle for > (greater than) and < (less than). For example: x > 1 = x is greater than 1
• Closed circle for ≥ (greater than or equal to) and ≤ (less than or equal to). For example: x ≤ 1 = x is less than or equal to 1
• Representing sets on a number line (e.g -2 < x ≤ 3) can be done like this:

The arrow on the number line points the same direction as the sign

Simulataneous Equations

it makes me want to eliminate… eliminate YOU!

You can solve simultaneous equations in different ways, graphically, using substitution, and using elimination. We will only cover substitution and elimination here.

Substitution

Substitute one equation into the other, and solve.

Example

y = 2x - 4 [1]
4x - y = 6 [2]

Sub [1] -> [2]

4x - (2x - 4) = 6
4x - 2x + 4 = 6
2x + 4 = 6
2x = 2
x = 1

Elimination

You can add or subtract equations together to eliminate pronumerals.

• Decide whether you should add or subtract.
• If you can’t add or subtract, multiply the equations to get a common pronumeral (similar to fractions)
• Eliminate the pronumeral and solve

Example

3x - y = 4 [1]
5x + y = 4 [2]

# add the equations together

3x + 5x + (-y) + y = 8
8x = 8
Therefore, x = 1

What if it isn’t possible to eliminate a pronumeral in the equations’ current form?

Example

7x - 2y = 3 [1]
4x - 5y = 6 [2]

# multiply [1] by 4 and [2] by 7

28x - 8y = 12 [1]
28x - 35y = -42 [2]

# [1] - [2]

-8y - 35y = 12 - (-42)
-43y = 54

ax² = c

where:

• a is the coefficient;
• x is the base; and
• c is the product.

Example

x² = 25

∴ x = ±√25 # surd form
∴ x = ±5   # basic numeral

Point of Intersection

The point where two lines intersect each other