Linear Inequalities in One Variable

Learning Objectives

  • Use the addition and multiplication properties to solve algebraic inequalities
  • Express solutions to inequalities graphically, with interval notation, and as an inequality
  • Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions

Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.

A General Note: Properties of Inequalities

[latex]\begin{array}{ll}\text{Addition Property}\hfill& \text{If }a< b,\text{ then }a+c< b+c.\hfill \\ \hfill & \hfill \\ \text{Multiplication Property}\hfill & \text{If }a< b\text{ and }c> 0,\text{ then }ac< bc.\hfill \\ \hfill & \text{If }a< b\text{ and }c< 0,\text{ then }ac> bc.\hfill \end{array}[/latex]

These properties also apply to [latex]a\le b[/latex], [latex]a>b[/latex], and [latex]a\ge b[/latex].

The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:

Start With Multiply By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex]ac>bc[/latex]
[latex]5>3[/latex] [latex]3[/latex] [latex]15>9[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex]-ac<-bc[/latex]
[latex]5>3[/latex] [latex]-3[/latex] [latex]-15<-9[/latex]

The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:

Start With Divide By Final Inequality
[latex]a>b[/latex] [latex]c[/latex] [latex]\displaystyle \frac{a}{c}>\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]2[/latex] [latex]\displaystyle \frac{4}{2}>\frac{2}{2}[/latex]
[latex]a>b[/latex] [latex]-c[/latex] [latex]\displaystyle -\frac{a}{c}<-\frac{b}{c}[/latex]
[latex]4>2[/latex] [latex]-2[/latex] [latex]\displaystyle -\frac{4}{2}<-\frac{2}{2}[/latex]

In the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.

Example: Demonstrating the Addition Property

Illustrate the addition property for inequalities by solving each of the following:

  1. [latex]x - 15<4[/latex]
  2. [latex]6\ge x - 1[/latex]
  3. [latex]x+7>9[/latex]

Try It

Solve [latex]3x - 2<1[/latex].

Example: Demonstrating the Multiplication Property

Illustrate the multiplication property for inequalities by solving each of the following:

  1. [latex]3x<6[/latex]
  2. [latex]-2x - 1\ge 5[/latex]
  3. [latex]5-x>10[/latex]

Try It

Solve [latex]4x+7\ge 2x - 3[/latex].

Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

Example: Solving an Inequality Algebraically

Solve the inequality: [latex]13 - 7x\ge 10x - 4[/latex].

Try It

Solve the inequality and write the answer using interval notation: [latex]-x+4<\frac{1}{2}x+1[/latex].

Example: Solving an Inequality with Fractions

Solve the following inequality and write the answer in interval notation: [latex]-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x[/latex].

Try It

Solve the inequality and write the answer in interval notation: [latex]-\frac{5}{6}x\le \frac{3}{4}+\frac{8}{3}x[/latex].

Simplify and solve algebraic inequalities using the distributive property

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

Example

Solve for x. [latex]2\left(3x–5\right)\leq 4x+6[/latex]

Check the solution.

In the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.

Try It