Inequalities
Symbols
● < → less than
● > → greater than
● ≤ → less than or equal to
● ≥ → greater than or equal to
● Number line inequalities - filled-in circle means ‘or equal to’
● Graphical inequalities - solid line means ‘or equal to’
Linear Inequalities
● Involves only x terms and constants
● Solve linear inequalities as you would solve an algebraic equation for the
most part
● However if you multiply or divide each side by a negative number the
inequality symbol must be reversed
● Check solutions by picking a number and plugging it into the inequality to
see if it satisfies your solution
Example
S olve 1 − x ≥ 2x − 5
1 − x ≥ 2x − 5
1 ≥ 3x − 5
6 ≥ 3x
x ≤2
Quadratic Inequalities
● Quadratic inequalities solved by factorising quadratic expression
● The numbers in the brackets tell you the boundaries of the solutions
● To decide which inequality signs to use, visualise graph
Example
S olve 3 − 5x ≤ 2x2
3 − 5x − 2x2 ≤ 0
2x2 + 5x − 3 ≥ 0
(2x − 1)(x + 3) ≥ 0
x ≥ 21 or x ≤ −3
Symbols
● < → less than
● > → greater than
● ≤ → less than or equal to
● ≥ → greater than or equal to
● Number line inequalities - filled-in circle means ‘or equal to’
● Graphical inequalities - solid line means ‘or equal to’
Linear Inequalities
● Involves only x terms and constants
● Solve linear inequalities as you would solve an algebraic equation for the
most part
● However if you multiply or divide each side by a negative number the
inequality symbol must be reversed
● Check solutions by picking a number and plugging it into the inequality to
see if it satisfies your solution
Example
S olve 1 − x ≥ 2x − 5
1 − x ≥ 2x − 5
1 ≥ 3x − 5
6 ≥ 3x
x ≤2
Quadratic Inequalities
● Quadratic inequalities solved by factorising quadratic expression
● The numbers in the brackets tell you the boundaries of the solutions
● To decide which inequality signs to use, visualise graph
Example
S olve 3 − 5x ≤ 2x2
3 − 5x − 2x2 ≤ 0
2x2 + 5x − 3 ≥ 0
(2x − 1)(x + 3) ≥ 0
x ≥ 21 or x ≤ −3
