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# Solving Inequalities

### Learning Outcomes

• Solve single-step inequalities
• Solve multi-step inequalities

## Multiplication and Division Properties of Inequality

Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. 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 similar to but not exactly as we treat equations. 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. 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 $a>b$ $c$ $ac>bc$ $5>3$ $3$ $15>9$ $a>b$ $-c$ $-ac<-bc$ $5>3$ $-3$ $-15<-9$
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 $a>b$ $c$ $\displaystyle \frac{a}{c}>\frac{b}{c}$ $4>2$ $2$ $\displaystyle \frac{4}{2}>\frac{2}{2}$ $a>b$ $-c$ $\displaystyle -\frac{a}{c}<-\frac{b}{c}$ $4>2$ $-2$ $\displaystyle -\frac{4}{2}<-\frac{2}{2}$
In the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.

### Example

Illustrate the multiplication property for inequalities by solving each of the following:
1. $3x<6$
2. $-2x - 1\ge 5$
3. $5-x>10$

Answer: a. [latex-display]\begin{array}{cc}\hfill3x<6 \hfill\\\dfrac{1}{3}\normalsize\left(3x\right)<\left(6\right)\dfrac{1}{3} \\ \hfill{x}<2 \hfill\end{array}[/latex-display]   b. [latex-display]\begin{array}{rr}-2x - 1\ge 5\\ \hfill\hfill-2x\ge 6\end{array}[/latex-display] Multiply both sides by $-\dfrac{1}{2}$. [latex-display]\begin{array}{ll}\hfill\hfill\left(-\dfrac{1}{2}\normalsize\right)\left(-2x\right)\ge \left(6\right)\left(-\dfrac{1}{2}\normalsize\right)\end{array}[/latex-display] Reverse the inequality. [latex-display]\begin{array}{l}\hfill&\hfill&\hfill&\hfill&\hfill x\le -3\end{array}[/latex-display] c. [latex-display]\begin{array}{ll}5-x>10\\ -x>5\hfill &\hfill\end{array}[/latex-display] Multiply both sides by $-1$. [latex-display]\left(-1\right)\left(-x\right)>\left(5\right)\left(-1\right)[/latex-display] Reverse the inequality [latex-display]x<-5[/latex-display]

## Solve Inequalities Using the Addition Property

When we solve equations, we may need to add or subtract in order to isolate the variable; the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities. The following table illustrates how the addition property applies to inequalities.
 Start With Add Final Inequality $a>b$ $c$ $a+c>b+c$ $5>3$ $3$ $8>6$ $a>b$ $-c$ $a-c>b-c$ $5>3$ $-3$ $2>0$
These properties also apply to $a\le b$, $a>b$, and $a\ge b$. In our next example, we will use the addition property to solve inequalities.

### Example

Illustrate the addition property for inequalities by solving each of the following:
1. $x - 15<4$
2. $6\ge x - 1$
3. $x+7>9$

Answer: The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality. a. [latex-display]\begin{array}{rr}\hfill x - 15<4\hfill\hfill \\ \hfill x - 15+15<4+15\hfill& \text{Add 15 to both sides.}\hfill\\\hfill\quad x<19 \hfill\end{array}[/latex-display] b. [latex-display]\begin{array}{rr}\hfill 6≥ x - 1\hfill\hfill \\\hfill 6+1\ge x - 1+1\hfill & \text{Add 1 to both sides}.\hfill \\\quad\quad 7≥ x\hfill \end{array}[/latex-display] c. [latex-display]\begin{array}{rr}\hfill x+7>9\hfill\hfill\\\hfill x+7 - 7>9 - 7\hfill & \text{Subtract 7 from both sides}.\hfill\quad \\\hfill x>2\hfill \end{array}[/latex-display]

The following video shows examples of solving single-step inequalities using the multiplication and addition properties. [embed]https://youtu.be/1Z22Xh66VFM[/embed] The following video shows examples of solving inequalities with the variable on the right side. [embed]https://youtu.be/RBonYKvTCLU[/embed] [ohm_question]77757[/ohm_question]

## Solve 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. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.

### Example

Solve the inequality: $13 - 7x\ge 10x - 4$.

Answer: Solving this inequality is similar to solving an equation up until the last step.

$\begin{array}{rr}13 - 7x\ge 10x - 4\hfill & \\ 13 - 17x\ge -4\hfill & \text{Move variable terms to one side of the inequality}.\hfill&\quad \\-17x\ge -17\hfill&\text{Isolate the variable term}.\hfill&\quad \\x\le 1\hfill & \text{Dividing both sides by -17 reverses the inequality}.\hfill \end{array}$
The solution set is given by the interval $\left(-\infty ,1\right]$, or all real numbers less than and including 1.

### Try It

[ohm_question]143594[/ohm_question]
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.

### Example

Solve the following inequality and write the answer in interval notation: $-\dfrac{3}{4}\normalsize x\ge -\dfrac{5}{8}\normalsize +\dfrac{2}{3}\normalsize x$.

Answer: We begin solving in the same way we do when solving an equation.

$\begin{array}{rr}-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x\hfill & \hfill \\ -\frac{3}{4}x-\frac{2}{3}x\ge -\frac{5}{8}\hfill & \text{Put variable terms on one side}.\hfill \\ -\frac{9}{12}x-\frac{8}{12}x\ge -\frac{5}{8}\hfill & \text{Write fractions with common denominator}.\hfill \\ -\frac{17}{12}x\ge -\frac{5}{8}\hfill & \hfill \\ x\le -\frac{5}{8}\left(-\frac{12}{17}\right)\hfill & \text{Multiplying by a negative number reverses the inequality}.\hfill \\ x\le \frac{15}{34}\hfill & \hfill \end{array}$
The solution set is the interval $\left(-\infty ,\dfrac{15}{34}\normalsize\right]$.

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