Simplify Inequalities w/ Distributive Property - Examples

The general approach to solve a complex inequality is the same as a simpler one. We will simplify and isolate our variable on one side of the equation. Let's focus on a technique based on the distributive property.

Remember, when you're working with inequalities you must FLIP the inequality sign if you multiply or divide both sides by a negative number.

Let's look at another example:

$$3(x - 2) > 7 + 5x$$

You can usually tell when you need to use the distributive property when you see an expression in parentheses (in this case, $x-2$) with a coefficient outside. We saw this before with simplifying equations.

As a reminder, think of "distributing" a value like dragging a paintbrush through the parentheses. We're going to be pulling the $3$ (in this case) across each of the terms in the parentheses.

$$3x - 6 > 7 + 5x$$

Now that we've handled the distribution, we can isolate $x$ on one side of the equation and solve.

$$\begin{align} 3x - 6 &> 7 + 5x \\ -2x - 6 &> 7 \\ -2x &> 13 \\\ \\ x &< -\frac{13}{2} \\ \end{align}$$

Notice that we flipped the inequality sign from "greater than" to "less than" because we divided both sides of the expression by $-2$.

(Video) Solve Single Linear Inequalities

by HEGARTYMATHS

Here's a good video by HEGARTYMATHS solving some inequalities using the distributive property.

$\large{\text{Summary}}$

Just like when solving normal equations, the distributive property means that you have to multiply each term inside the parentheses by the term immediately outside the parentheses. In the expression

$$4(3x - 2)$$

That means that you need to multiply $3x$ by $4$ and $2$ by $4$. This will give you:

$$12x - 8$$

After distributing the multiplication, it doesn't hurt to also combine like terms to clean up your equation.

$$\begin{align} 2x + 3(4x - 3) &> 2(-x + 1) -3(9 + 7x)\\ 2x + 12x - 9 &> -2x + 2 - 27 -21x)\\\ \\ 14x - 9 &> - 25 - 23x \end{align}$$

Simplify Inequalities: Using the Distributive Property

Solving inequalities with the distributive property is very similar to solving regular equations with the distributive property. There are three simple steps to follow to solve the inequality.

Try an example problem!

The solution: