Writing inequalities

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:

• [latex]{x}\lt{9}[/latex] indicates the list of numbers that are less than 9. Would you rather write [latex]{x}\lt{9}[/latex] or try to list all the possible numbers that are less than 9? (hopefully, your answer is “no”, you don’t want to try to list them all.)
• [latex]-5\le{t}[/latex] indicates all the numbers that are greater than or equal to [latex]-5[/latex].

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:

• [latex]x\lt5[/latex] means all the real numbers that are less than 5, whereas;
• [latex]5\lt{x}[/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\gt{5}[/latex]. 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:

The third way is with an interval.

The fourth way is using set-builder notation.

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

• an inequality
• a graph
• an interval
• using set-builder notation

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.

The inequality [latex]x>y[/latex] can also be written as [latex]{y}<{x}[/latex]. 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!

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

[latex]{x}\geq{-3}[/latex]. 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 [latex] \displaystyle \left(\geq\right) [/latex] or less than or equal to [latex] \displaystyle \left(\leq\right) [/latex]. 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 [latex] \displaystyle x\geq -3[/latex] shown above, the end point is [latex]−3[/latex], represented with a closed circle since the inequality is greater than or equal to [latex]−3[/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]−3[/latex]. The arrow at the end indicates that the solutions continue infinitely.

Example 1

Graph the inequality [latex]x\ge 4[/latex].

Show Solution

We can use a number line as shown. Because the values for x include 4, we place a solid dot on the number line at 4.

Then we draw a line that begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.

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

Example 2

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

Show Solution

We need to start from the left and work right, so we start from negative infinity and end at [latex]-2[/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]-2[/latex].

Inequality: [latex]x<2[/latex]

To draw the graph, place an open dot on the number line first, then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.

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 [latex]x\geq 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. 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 [latex]\left[-2,6\right)[/latex], or all numbers between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/latex], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers between, but not including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty,1\right][/latex], all real numbers less than and including [latex]1[/latex]. To construct intervals, start with the value that is furthest left on the number line as the left value, such as [latex]\left[-2,6\right)[/latex], where -2 is less than 6. The number on the right should be greater than the number on the left. While not technically numbers, this ordering extends to positive and negative infinity as well.

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.  Remember to read inequalities from left to right, just like text.

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

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.

Think About It 1

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 [latex]\left(-\infty,10\right)[/latex], 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.

Show Solution

We will draw the graph first.

The interval reads “all real numbers less than 10,” so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity.

To write the inequality, we will use < since the parentheses indicate that 10 is not included. [latex]x<10[/latex]

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.

Represent inequalities using set-builder notation

Another method used for describing inequalities and solutions to inequalities is called set-builder notation.  In this method, we show the conditions a variable must meet in order to be part of a set.

If we want to describe the set of all Real numbers that are greater than 0 using set-builder notation, it would look like the following:

[latex]{\color{red}\{}{\color{green}x}\hspace{.01in}{\mathbf{\color{blue}|}}\hspace{.01in}x>0{\color{red}\}}[/latex]

The parentheses indicate that we have a set. The vertical line, |, is read “such that”. Therefore, the example above would read as:

“The set of all x such that x is greater than 0.”

Let’s look at one more example.  Let’s use set-builder notation to describe the set of all Real numbers that are between 4 and 10, including 10.

[latex]{\color{red}\{}{\color{green}x}\hspace{.01in}{\mathbf{\color{blue}|}}\hspace{.01in}4<x\le10{\color{red}\}}[/latex]

The set of all x such that x is between 4 and 10, including 10.

Solving 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.

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.

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 [latex]x<2[/latex] is the solution to [latex]x+3<5[/latex].

Example 8

Check that [latex]x<2[/latex] is the solution to [latex]x+3<5[/latex].

Show Solution

Substitute the end point 2 into the related equation, [latex]x+3=5[/latex].

[latex]\begin{array}{r}x+3=5 \\ 2+3=5 \\ 5=5\end{array}[/latex]

Pick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)

[latex] \displaystyle \begin{array}{r}x+3<5 \\ 0+3<5 \\ 3<5\end{array}[/latex]

It checks!

[latex]x<2[/latex] is the solution to [latex]x+3<5[/latex].

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 10

Solve for a: [latex]\hspace{.05in}a-17>-17[/latex]

Show Solution

Isolate the variable by adding 17 to both sides of the inequality.

[latex] \displaystyle \begin{array}{r}a-17>-17\\\underline{\,\,\,+17\,\,\,\,\,+17}\\a\,\,\,\,\,\,\,\,\,\,\,>\,\,\,\,\,\,0\end{array}[/latex]

Answer

Inequality: [latex] \displaystyle a\,\,>\,0[/latex]

Interval: [latex]\left(0,\infty\right)[/latex]  Note how we use parentheses on the left to show that the solution does not include 0.

Graph: Note the open circle to show that the solution does not include 0.

Check the solution to [latex]a-17>-17[/latex]

Show Solution

Is [latex] \displaystyle a\,\,>\,0[/latex] the correct solution to  [latex]a-17>-17[/latex]?

Substitute the end point 0 into the related equation.

[latex] \displaystyle \begin{array}{r}a-17=-17\,\,\,\\\text{Does}\,\,\,0-17=-17?\\-17=-17\,\,\,\end{array}[/latex]

Pick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)

[latex] \displaystyle \begin{array}{r}a-17>-17\,\,\,\\\text{Is }\,\,20-17>-17?\\3>-17\,\,\,\\\\\text{It checks!}\,\,\,\,\end{array}[/latex]

[latex] \displaystyle a\,>\,0[/latex] is the solution to [latex]a-17>-17[/latex]

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.

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.

Remember!  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.

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.

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!

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.

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.

Combining 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 14

Solve for p: [latex]\hspace{.05in}4p+5<29[/latex]

Show Solution

Begin to isolate the variable by subtracting 5 from both sides of the inequality.

[latex] \displaystyle \begin{array}{l}4p+5<\,\,\,29\\\underline{\,\,\,\,\,\,\,\,\,-5\,\,\,\,\,-5}\\4p\,\,\,\,\,\,\,\,\,<\,\,24\,\,\end{array}[/latex]

Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.

[latex]\begin{array}{l}\underline{4p}\,<\,\,\underline{24}\,\,\\\,4\,\,\,\,<\,\,4\\\,\,\,\,\,p<6\end{array}[/latex]

Answer

Inequality: [latex]p<6[/latex]

Interval: [latex]\left(-\infty,6\right)[/latex]

Graph: Note the open circle at the end point 6 to show that solutions to the inequality do not include 6. The values where p is less than 6 are found all along the number line to the left of 6.

Check the solution.

Show Solution

Check the end point 6 in the related equation.

[latex] \displaystyle \begin{array}{r}4p+5=29\,\,\,\\\text{Does}\,\,\,4(6)+5=29?\\24+5=29\,\,\,\\29=29\,\,\,\\\text{Yes!}\,\,\,\,\,\,\end{array}[/latex]

Try another value to check the inequality. Let’s use [latex]p=0[/latex].

[latex] \displaystyle \begin{array}{r}4p+5<29\,\,\,\\\text{Is}\,\,\,4(0)+5<29?\\0+5<29\,\,\,\\5<29\,\,\,\\\text{Yes!}\,\,\,\,\,\end{array}[/latex]

[latex]p<6[/latex] is the solution to [latex]4p+5<29[/latex]

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 16

Solve for p: [latex]\hspace{.05in}−58>14−6p[/latex]

Show Solution

Note how the variable is on the right hand side of the inequality, the method for solving does not change in this case.

Begin to isolate the variable by subtracting 14 from both sides of the inequality.

[latex] \displaystyle \begin{array}{l}−58\,\,>14−6p\\\underline{-14\,\,\,-14}\\-72\,\,>-6p\end{array}[/latex]

Divide both sides of the inequality by [latex]−6[/latex] to express the variable with a coefficient of 1. Dividing by a negative number results in reversing the inequality sign.

[latex]\begin{array}{l}\underline{-72}>\underline{-6p}\\-6\,\,\,\,\,\,\,\,\,\,-6\\\,\,\,\,\,\,12\lt{p}\end{array}[/latex]

We can also write this as [latex]p>12[/latex].   Notice how the inequality sign is still opening up toward the variable p.

Answer

Inequality: [latex]p>12[/latex]
Interval: [latex]\left(12,\infty\right)[/latex]
Graph: The graph of the inequality p > 12 has an open circle at 12 with an arrow stretching to the right.

Check the solution.

Show Solution

First, check the end point 12 in the related equation.

[latex]\begin{array}{r}-58=14-6p\\-58=14-6\left(12\right)\\-58=14-72\\-58=-58\end{array}[/latex]

Then, try another value to check the inequality. Try 100.

[latex]\begin{array}{r}-58>14-6p\\-58>14-6\left(100\right)\\-58>14-600\\-58>-586\end{array}[/latex]

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.

Simplifying and solving 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 17

Solve for x: [latex]\hspace{.05in}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.

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 four ways to represent solutions to inequalities: an inequality, a graph, an interval, and using set-builder notation. 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.