Properties of Inequality

A linear inequality is similar to a linear equation in many ways, but do the properties of equality still hold? An inequality like [latex]2\lt 5[/latex] can be visualized as an unbalanced scale:

It seems logical that adding or subtracting the same amount from both sides of the scale will not change the imbalance.

Consider the simple inequality [latex]2 \lt 5[/latex]. Let’s add [latex]4[/latex] to each side and determine if the inequality still holds:

[latex]\begin{equation}\begin{aligned}2 & \lt\ 5 \\ 2 \color{blue}{+4} & \lt 5\color{blue}{+4} \\ 6 & \lt 9 \end{aligned}\end{equation}[/latex]

Adding the same number to both sides keeps the inequality in tact. Let’s try subtracting [latex]3[/latex] from both sides to see what happens:

[latex]\begin{equation}\begin{aligned}2 & \lt\ 5 \\ 2 \color{blue}{-3} & \lt 5\color{blue}{-3} \\ -1 & \lt 2 \end{aligned}\end{equation}[/latex]

Subtracting the same number to both sides keeps the inequality in tact.

The addition and subtraction property of equality applies to inequalities. But what of the multiplication and division properties of equality? Let’s consider the inequality [latex]2\lt 5[/latex] again, and start by multiplying both sides by a positive number.

[latex]\begin{equation}\begin{aligned}2 & \lt\ 5 \\ 2 \color{blue}{\cdot 3} & \lt 5\color{blue}{\cdot 3} \\ 6 & \lt 15 \end{aligned}\end{equation}[/latex]

The inequality still holds. But what about multiplying by a negative number?

[latex]\begin{equation}\begin{aligned}2 & \lt\ 5 \\ 2 \color{blue}{\cdot (-4)} & \lt 5\color{blue}{\cdot (-4)} \\ -8 & \lt -20 \end{aligned}\end{equation}[/latex]

Thuis results in a false statement! To make the statement true, we must flip the [latex]\lt[/latex] sign to a [latex]\gt[/latex] sign because [latex]-8\gt -20[/latex]. The same thing happens if we try to divide by a negative number: we must flip the inequality sign.

Solving an linear inequality follows the same rules as solving a linear equation. In the first example, we use the multiplication and division property to isolate the variable.

In our next example, we will use the addition property to solve inequalities.

The following video shows examples of solving single-step inequalities using the multiplication and addition properties.

The following video shows examples of solving inequalities with the variable on the right side.

Solving Multi-Step Inequalities

As the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. We just have to remember to reverse the sign if we multiply or divide by a negative term. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.

In the next example, we solve an inequality that contains fractions. Notice how we need to reverse the inequality sign at the end because we multiply by a negative.

Instead of working with the fractions in this last example, we could clear the fractions by multiplying both sides by a common denominator.