Equations and Inequalities | Year 9 Mathematics (Cambridge) PDF Download

Linear Equations

Introduction to Linear Equations

A linear equation is an equation that, when graphed, forms a straight line. The standard form of a linear equation is:
ax + b = 0
where 'a' and 'b' are constants, and 'x' is the variable.

Solving Linear Equations

To solve a linear equation, follow these steps to isolate the variable 'x':

• Simplify both sides: Combine like terms on both sides of the equation, if any.
• Move the variable term to one side: Use addition or subtraction to place the variable term on one side and the constant term on the other.
• Isolate the variable: Divide or multiply both sides by the coefficient of the variable to find the value of 'x'.

Examples

Example 1: Solve 3x + 5 = 11
Solution: 3x + 5 = 11
Subtract 5 from both sides:
3x = 6
Divide both sides by 3:
x = 2

Example 2: Solve 2x − 4 = 10
Solution: 2x − 4 = 10
Add 4 to both sides:
2x = 14
Divide both sides by 2:
x = 7

Example 3: Solve 5x + 2 = 3x + 10
Solution: 5x + 2 = 3x + 10
Subtract 3x from both sides:
2x + 2 = 10
Subtract 2 from both sides:
2x = 8
Divide both sides by 2:
x = 4

Solving Inequalities

Introduction to Inequalities

An inequality is like an equation, but instead of an equals sign (=), it uses inequality symbols:

• < (less than)
• ≤ (less than or equal to)
• > (greater than)
• ≥ (greater than or equal to)

Solving Inequalities

The process of solving inequalities is similar to solving linear equations, but there are some important differences:

• Simplify both sides: Combine like terms on each side of the inequality.
• Move the variable term to one side: Use addition or subtraction to get the variable term on one side and the constant term on the other.
• Isolate the variable: Divide or multiply both sides by the coefficient of the variable. Important: If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.

Examples

Example 1: Solve 2x + 3 < 7
Solution: 2x + 3 < 7
Subtract 3 from both sides:
2x < 4
Divide both sides by 2:
x < 2

Example 2: Solve 4x − 5 ≥ 3
Solution: 4x − 5 ≥ 3
Add 5 to both sides:
4x ≥ 8
Divide both sides by 4:
x ≥ 2

Example 3: Solve −3x + 6 > 0
Solution: −3x + 6 > 0
Subtract 6 from both sides:
−3x > −6
Divide both sides by -3 (remember to reverse the inequality):
x < 2

Graphing Solutions

Graphing Linear Equations

When graphing a linear equation y = mx + b:

• m is the slope (rise over run).
• b is the y-intercept (where the line crosses the y-axis).

Graphing Inequalities

When graphing inequalities, the graph will show a region that satisfies the inequality:

• Use a dashed line for < or > (the boundary is not included).
• Use a solid line for ≤ or ≥ (the boundary is included).
• Shade the region where the inequality holds true.

Example

Graph the inequality y ≤ 2x + 1:

• Graph the line y = 2x + 1 with a solid line.
• Shade the region below the line (because y is less than or equal to).

Summary

• Linear equations form a straight line and can be solved by isolating the variable.
• Inequalities represent a range of values and require careful handling, especially when multiplying or dividing by negative numbers.
• Graphing solutions helps visualize the set of possible solutions for both equations and inequalities.