Algebraic Inequalities

An inequality expresses a relationship between two expressions, not equality, but a magnitude relation. The most common signs are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

In this inequality, we are looking for what x values make 2x + 1 greater than 7.

Basic Solution Steps

• We can perform the same operations on both sides as with equations: addition, subtraction, multiplication, division.
• Important difference: if we multiply or divide by a negative number, the inequality direction reverses.
• The solution is often not a single number, but an interval or multiple possible value sets.

Example: Simple Inequality

Subtract 1 from both sides: 2x > 6. Divide by 2: x > 3. So every number greater than 3 is a solution.

Multiplying or Dividing by Negative

If we multiply or divide by a negative number, the inequality direction reverses. This is the most important rule that differs from solving equations.

Divide by -3: x > -3. Notice that the < sign changed to >!

Representing Inequality Solutions

Inequalities are often represented on a number line. An open circle (○) indicates that the endpoint is not included (< or >), a closed circle (●) indicates it is included (≤ or ≥).

Everyday Examples

• If a concert ticket is for those 18 and older, the condition is: age ≥ 18.
• If a running race has a time limit of 60 minutes, the condition is: time ≤ 60.
• If a store offers free shipping only for orders over 10,000 Ft, the condition is: amount > 10000.

Practice Exercise