## Solve Inequalities

### Learning Outcomes

• Describe solutions to inequalities
• Represent inequalities on a number line
• Represent inequalities using interval notation
• Solve single-step inequalities
• Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation
• Solve inequalities that contain absolute value
• Solve multi-step inequalities
• Combine properties of inequality to isolate variables, solve algebraic inequalities, and express their solutions graphically
• Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions

## Represent inequalities on a number line

First, let’s define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right—just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.

The first way you are probably familiar with—the basic inequality. For example:

• ${x}\lt{9}$ indicates the list of numbers that are less than 9. Would you rather write ${x}\lt{9}$ or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)
• $-5\le{t}$ indicates all the numbers that are greater than or equal to $-5$.

Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.

For example:

• $x\lt5$ means all the real numbers that are less than 5, whereas;
• $5\lt{x}$ means that 5 is less than x, or we could rewrite this with the x on the left: $x\gt{5}$ note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than x.

The second way is with a graph using the number line:

And the third way is with an interval.

We will explore the second and third ways in depth in this section. Again, those three ways to write solutions to inequalities are:

• an inequality
• an interval
• a graph

### Inequality Signs

The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it’s easy to get tangled up in inequalities, just remember to read them from left to right.

Symbol Words Example
$\neq$ not equal to ${2}\neq{8}$, 2 is not equal to 8.
$\gt$ greater than ${5}\gt{1}$, 5 is greater than 1
$\lt$ less than ${2}\lt{11}$, 2 is less than 11
$\geq$ greater than or equal to ${4}\geq{ 4}$, 4 is greater than or equal to 4
$\leq$ less than or equal to ${7}\leq{9}$, 7 is less than or equal to 9

The inequality $x>y$ can also be written as ${y}<{x}$. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

## Graphing an Inequality

Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs.  Graphs are a very helpful way to visualize information – especially when that information represents an infinite list of numbers!

$x\leq -4$. This translates to all the real numbers on a number line that are less than or equal to 4.

${x}\geq{-3}$. This translates to all the real numbers on the number line that are greater than or equal to -3.

Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to $\displaystyle \left(\geq\right)$ or less than or equal to $\displaystyle \left(\leq\right)$. The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution.

The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of $\displaystyle x\geq -3$ shown above, the end point is $−3$, represented with a closed circle since the inequality is greater than or equal to $−3$. The blue line is drawn to the right on the number line because the values in this area are greater than $−3$. The arrow at the end indicates that the solutions continue infinitely.

### Example

Graph the inequality $x\ge 4$

This video shows an example of how to draw the graph of an inequality.

### Example

Write an inequality describing all the real numbers on the number line that are less than 2, then draw the corresponding graph.

The following video shows how to write an inequality mathematically when it is given in words. We will then graph it.

## Represent inequalities using interval notation

Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called interval notation. With this convention, sets are built with parentheses or brackets, each having a distinct meaning. The solutions to $x\geq 4$ are represented as $\left[4,\infty \right)$. This method is widely used and will be present in other math courses you may take.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are $\left[-2,6\right)$, or all numbers between $-2$ and $6$, including $-2$, but not including $6$; $\left(-1,0\right)$, all real numbers between, but not including $-1$ and $0$; and $\left(-\infty,1\right]$, all real numbers less than and including $1$. The table below outlines the possibilities. Remember to read inequalities from left to right, just like text.

The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.

Inequality Words Interval Notation
${a}\lt{x}\lt{ b}$ all real numbers between a and b, not including a and b $\left(a,b\right)$
${x}\gt{a}$ All real numbers greater than a, but not including a $\left(a,\infty \right)$
${x}\lt{b}$ All real numbers less than b, but not including b $\left(-\infty ,b\right)$
${x}\ge{a}$ All real numbers greater than a, including a $\left[a,\infty \right)$
${x}\le{b}$ All real numbers less than b, including b $\left(-\infty ,b\right]$
${a}\le{x}\lt{ b}$ All real numbers between a and b, including a $\left[a,b\right)$
${a}\lt{x}\le{ b}$ All real numbers between a and b, including b $\left(a,b\right]$
${a}\le{x}\le{ b}$ All real numbers between a and b, including a and b $\left[a,b\right]$
${x}\lt{a}\text{ or }{x}\gt{ b}$ All real numbers less than a or greater than b $\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers All real numbers $\left(-\infty ,\infty \right)$

### Example

Describe the inequality $x\ge 4$ using interval notation

In the following video we show another example of using interval notation to describe an inequality.

### Example

Use interval notation to indicate all real numbers greater than or equal to $-2$.

In the following video we show another example of translating words into an inequality and writing it in interval notation, as well as drawing the graph.

In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph.

Given $\left(-\infty,10\right)$, write the associated inequality and draw the graph.

In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.

In the following video, you will see examples of how to draw a graph given an inequality in interval notation.

And finally, one last video that shows how to write inequalities using a graph, with interval notation and as an inequality.

## Solve Single-Step Inequalities

### Solve inequalities with addition and subtraction

You can solve most inequalities using inverse operations  as you did for solving equations.  This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.

### Addition and Subtraction Properties of Inequality

If $a>b$, then $a+c>b+c$.

If $a>b$, then $a−c>b−c$.

Because inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.

### Example

Solve for x.

${x}+3\lt{5}$

The line represents all the numbers to which you can add 3 and get a number that is less than 5. There’s a lot of numbers that solve this inequality!

Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.

The example below shows how you could check that $x<2$ is the solution to $x+3<5$.

### Example

Check that $x<2$ is the solution to $x+3<5$.

The following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!

### Example

Solve for x: $x-10\leq-12$

Check the solution to $x-10\leq -12$

### Example

Solve for a. $a-17>-17$

Check the solution to $a-17>-17$

The previous examples showed you how to solve a one-step inequality with the variable on the left hand side.  The following video provides examples of how to solve the same type of inequality.

What would you do if the variable were on the right side of the inequality?  In the following example, you will see how to handle this scenario.

### Example

Solve for x: $4\geq{x}+5$

Check the solution to $4\geq{x}+5$

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

### Solve inequalities with multiplication and division

Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let’s look at what happens to the inequality when you multiply or divide each side by the same number.

 Let’s start with the true statement: $10>5$ Let’s try again by starting with the same true statement: $10>5$ Next, multiply both sides by the same positive number: $10\cdot 2>5\cdot 2$ This time, multiply both sides by the same negative number: $10\cdot-2>5 \\ \,\,\,\,\,\cdot -2\,\cdot-2$ 20 is greater than 10, so you still have a true inequality: $20>10$ Wait a minute! $−20$ is not greater than $−10$, so you have an untrue statement. $−20>−10$ When you multiply by a positive number, leave the inequality sign as it is! You must “reverse” the inequality sign to make the statement true: $−20<−10$

Caution!  When you multiply or divide by a negative number, “reverse” the inequality sign.   Whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.

### Multiplication and Division Properties of Inequality

 Start With Multiply By Final Inequality $a>b$ $c$ $ac>bc$ $a>b$ $-c$ $ac  Start With Divide By Final Inequality [latex]a>b$ $c$ $\displaystyle \frac{a}{c}>\frac{b}{c}$ $a>b$ $-c$ $\displaystyle \frac{a}{c}<\frac{b}{c}$

Keep in mind that you only change the sign when you are multiplying and dividing by a negative number. If you add or subtract by a negative number, the inequality stays the same.

### Example

Solve for x. $3x>12$

There was no need to make any changes to the inequality sign because both sides of the inequality were divided by positive 3. In the next example, there is division by a negative number, so there is an additional step in the solution!

### Example

Solve for x. $−2x>6$

The following video shows examples of solving one step inequalities using the multiplication property of equality where the variable is on the left hand side.

Before you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.

Solve for x. $-\frac{1}{2}>-12x$

The following video gives examples of how to solve an inequality with the multiplication property of equality where the variable is on the right hand side.

## Combine properties of inequality to  solve algebraic inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.

### Example

Solve for p. $4p+5<29$

Check the solution.

### Example

Solve for x:  $3x–7\ge 41$

Check the solution.

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

### Example

Solve for p. $−58>14−6p$

Check the solution.

In the following video, you will see an example of solving a linear inequality with the variable on the left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

In the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.

## 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. $2\left(3x–5\right)\leq 4x+6$

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.

In the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.

Solve for a. $\displaystyle\frac{{2}{a}-{4}}{{6}}{<2}$

Check the solution.

## Summary

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.

Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract either positive or negative numbers to both sides of the inequality.