(6.1.1) – 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.

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

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

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

(6.1.2) – Solve linear inequalities and express solutions with interval notation and as an inequality

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.

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

Show Solution

Distribute to clear the parentheses.

[latex] \displaystyle \begin{array}{r}\,2(3x-5)\leq 4x+6\\\,\,\,\,6x-10\leq 4x+6\end{array}[/latex]

Subtract 4x from both sides to get the variable term on one side only.

[latex]\begin{array}{r}6x-10\le 4x+6\\\underline{-4x\,\,\,\,\,\,\,\,\,\,\,\,\,\,-4x}\,\,\,\,\,\,\,\,\,\\\,\,\,2x-10\,\,\leq \,\,\,\,\,\,\,\,\,\,\,\,6\end{array}[/latex]

Add 10 to both sides to isolate the variable.

[latex]\begin{array}{r}\\\,\,\,2x-10\,\,\le \,\,\,\,\,\,\,\,6\,\,\,\\\underline{\,\,\,\,\,\,+10\,\,\,\,\,\,\,\,\,+10}\\\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\le \,\,\,\,\,16\,\,\,\end{array}[/latex]

Divide both sides by 2 to express the variable with a coefficient of 1.

[latex]\begin{array}{r}\underline{2x}\le \,\,\,\underline{16}\\\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\le \,\,\,\,\,8\end{array}[/latex]

Answer

Inequality: [latex]x\le8[/latex]
Interval: [latex]\left(-\infty,8\right][/latex]
Graph: The graph of this solution set includes 8 and everything left of 8 on the number line.

Check the solution.

Show Solution

First, check the end point 8 in the related equation.

[latex] \displaystyle \begin{array}{r}2(3x-5)=4x+6\,\,\,\,\,\,\\2(3\,\cdot \,8-5)=4\,\cdot \,8+6\\\,\,\,\,\,\,\,\,\,\,\,2(24-5)=32+6\,\,\,\,\,\,\\2(19)=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\38=38\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 0.

[latex] \displaystyle \begin{array}{l}2(3\,\cdot \,0-5)\le 4\,\cdot \,0+6?\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2(-5)\le 6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-10\le 6\,\,\end{array}[/latex]

[latex]x\le8[/latex] is the solution to [latex]\left(-\infty,8\right][/latex]

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