Inequalities

Understanding Inequalities

• An inequality shows the relationship between two expressions that may not be equal. It consists of two expressions separated by an inequality sign (> greater than, < less than, ≥ greater than or equal to, ≤ less than or equal to, ≠ not equal to).
• Solving inequalities is very similar to solving equations. The same methods can be used: addition, subtraction, multiplication, and division.
• When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

Working with Inequalities

• To solve an inequality like x + 3 > 7, subtract 3 from both sides to find the solution x > 4.
• To solve an inequality like -2x < 8, divide both sides by -2, and remember to reverse the inequality, resulting in x > -4.
• To represent an inequality on a number line, use an open circle for < and > situations, and a filled circle for ≤ and ≥ situations.

Inequalities and Interval Notation

• Interval notation is a method of writing down a range of numbers. For example, the inequality x ≥ 3 is represented as [3, ∞) in interval notation.
• The rounded bracket “)” means “up to but not including” and the square bracket “[” means “up to and including”.

Avoiding Common Mistakes

• Pay special attention when multiplying or dividing by negative numbers. The inequality sign must be swapped in these situations.
• Be sure to correctly represent inequalities on a number line, with the right notation for inclusive and exclusive limits.
• Do not confuse inequality signs: > is greater than, < is less than, ≥ is greater than or equal to, ≤ is less than or equal to, and ≠ is not equal to.