Algebraic Inequalities Study Guide

Introduction to Algebraic Inequalities

Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.

— Plato (C. 424/423— 348/347 B.C.) Classical Greek Philosopher

In this lesson, you'll learn how to solve single-variable and compound inequalities, and how to simplify algebraic inequalities with two variables.

If an equation is what we write to show two quantities that are equal to each other, what can we write to show that two quantities are NOT equal to each other? An inequality. Inequalities can use the less than sign (<), the greater than sign (>), the less than or equal to sign ≤, and the greater than or equal to sign ≥ to compare two quantities. An algebraic inequality is an algebraic expression that contains one of those four signs.

We solve single-variable equations by isolating the variable on one side of the equal sign and its value on the other side. We solve single-variable inequalities in the same way, except that instead of finding an answer that is a single value, our answer is a set of values.

The equation x + 4 = 9 is solved by subtracting 4 from both sides of the equal sign: x + 4 – 4 = 9 – 4, x = 5. The inequality x + 4 < 9 is solved in the same way: Subtract 4 from both sides of the less than sign: x + 4 – 4 < 9 – 4, x < 5. Our answer is x < 5, which means that all values of x that are less than 5 make the inequality true.

We can represent an inequality on a number line. To show x < 5, we put a circle around the number 5, because it is not a part of our answer (our solution is only values of x that are less than 5), and we highlight all of the values to the left of 5, to show that every number that is less than 5 is part of the solution:

There is one important difference between how we solve an equation and how we solve an inequality. When you are solving an inequality, if you multiply or divide both sides of the equation by a negative number, you must flip the inequality symbol. For example, to solve –5x < 25, divide both sides of the inequality by –5. When you do that, switch the inequality symbol from the less than sign to the greater than sign:

–5x < 25
x > 5

Why do we switch the symbol? Let's look at some real numbers. We know that –1 is less than 2, and we show that by writing –1 < 2. If we multiply both sides of the inequality by 2, we have –2 < 4, which is also true. The left side of the inequality became twice as small, and the right side of the inequality became twice as large. But what if we were to divide both sides of –1 < 2 by –1. The left side would become 1, and the right side would become –2. However, 1 is greater than –2. So, we must switch the less than sign to a greater than sign, to show that 1 > –2.

Let's look at another example: 3x – 7 ≥ 2. Add 7 to both sides and divide by 3:

3x – 7 ≥ 2
3x ≥ 9
x ≥ 3

The number line of this inequality shows a solid circle around 3, because 3 is part of the solution set (since 3 is greater than or equal to 3):