Linear Equations and Inequalities

Algebra and Sequences

Apply the same operations to both sides as when solving an equation
- 4x + 5 ⩽ 29 gives 4x ⩽ 24 and x ⩽ 6
A double inequality is solved by doing the same thing to all three parts
- −5 < 2x − 1 ⩽ 9 gives −4 < 2x ⩽ 10 and −2 < x ⩽ 5

A negative multiplier or divisor flips the direction of the inequality
- −3x > 12 gives x < −4, not x > −4
Avoiding the reversal altogether is often simpler: move the unknown to the side that keeps it positive

Worked example

Solving two inequalities

Solve 5x − 3 > 12, and solve −3x ⩾ 12.

Solution:

For the first, treat it exactly as an equation
Add 3 to both sides: 5x > 15
Divide both sides by 5: x > 3
The sign does not change, because 5 is positive
For the second, divide both sides by −3
Dividing by a negative reverses the inequality sign
12 ÷ −3 = −4, and ⩾ becomes ⩽, giving x ⩽ −4
Check with a value: x = −5 gives −3 × −5 = 15, and 15 ⩾ 12 is true
Leaving the sign as ⩾ would claim x ⩾ −4, and x = 0 gives 0 ⩾ 12, which is false