# 3.2: Inequalities on a Number Line

- Page ID
- 1088

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Inequalities on a Number Line

A scientific center has been conducting research about bird populations that are being harmed by human development. Scientists are carefully gathering information on shrinking habitats and gathering data about the biology of the birds affected. A group of student scientists went to visit the center and learned about the beak length of various birds. They discovered that the beak length was always 7 inches or less. Can you represent the range of possible beak lengths on a number line?

In this concept, you will learn to graph inequalities on a number line.

### Graphing Inequalities on a Number Line

An **inequality** is a mathematical statement that uses one of the following symbols: >, <, ≥, ≤, instead of an equals sign.

Here is what those symbols mean:

> greater than

< less than

≥ greater than or equal to

≤ less than or equal to

Let’s look at an example to see how to interpret these symbols.

If you have the statement x>2, what does it mean?

The inequality x>2 has a variable, x, that represents an infinite set of numbers. Although x cannot be equal to 2, it can be any real number greater than 2. It is impossible to write out every single number that x can be. For this reason, you will see that a number line is a useful tool to represent the solution. For now, let’s write out some of the numbers that are in the solution of x>2.

To make this statement true, x can be any of the following numbers.

{2.01,19/9,2.988,3,4,4.0000001,20/3,1000,1001.11111⋯}

You can start to see the range of numbers that make the statement x>2 true. Though it is helpful to see some of the numbers in the solution for x>2, it is best to represent the solution on a number line.

Let’s look at an example where you graph an inequality.

Graph the solution set for the inequality x>3 on a number line.

To help complete this task, first draw a number line from -5 to 5, marking off ticks at integer intervals.

The inequality x>3 is read as “x is greater than 3.” So the solution of this inequality includes all numbers greater than 3. It does * not*, however, actually include 3. To show that the answer does not include 3, you use an open circle.

Next, draw an arrow showing all numbers greater than 3. The arrow should point towards the right because greater numbers are to the right on a number line.

The answer is

Here are some tips for graphing inequalities on a number line.

Use an open circle to show that a value is * not* a solution for the inequality. You will use open circles to graph inequalities that include the symbols > or <.

Use a closed circle to show that a value * is* a solution for the inequality. You will use closed circles to graph inequalities that include the symbols ≥ or ≤.

Here is another example.

Graph the solution for x<-1 on a number line.

First, draw a number line from -5 to 5.

The inequality x<-1 is read as “x is less than -1.” The solutions of this inequality include all numbers less than -1, but not -1 itself, so draw an open circle at -1 to show that it is not a solution for this inequality. Then draw an arrow showing all numbers less than -1. The arrow should point towards the left because smaller numbers are to the left on a number line.

### Examples

Example 3.2.1

Earlier, you were given a problem about the students going to the scientific center to study birds and their shrinking environment.

**Solution**

These students are collecting data on beak length and found that all the birds they looked at had beaks 7 inches or less. They need to represent the range of beak lengths on a number line.

Since there were beak lengths that were 7 inches you use a solid circle on 7 inches. They also found beaks less than 7 inches. So you draw an arrow to the left.

One possible solution is the following.

This is a fine answer, but can you improve it?

A beak length cannot be less than zero, so a better solution would be a closed circle at 7, with a line going to the left, but then stopping at an open circle at 0. This would show that beak lengths were also always greater than 0.

Example 3.2.2

Graph the solution of x≥0 on a number line.

**Solution**

First, draw a number line from -5 to 5.

The inequality x≥0 is read as “x is greater than or equal to 0.” So, the answer includes zero and all numbers that are greater than 0.

Draw a closed circle at 0 to show that 0 is in the solution for this inequality. Then draw an arrow pointing to the right, showing all numbers greater than 0 are also part of the solution set.

The answer is the graph below which shows the solution for the inequality x≥0.

Example 3.2.3

True or false: An open circle on a number line means that the circled number is not included in the solution set.

Open circles are a way to indicate that the circled number is not part of the solution.

**Solution**

The answer is true.

Example 3.2.4

True or false: An inequality does not include the number referenced in the solution set.

Inequalities include greater than or equal to (≥) and less than or equal to (≤). When these symbols are used the number referenced is included in the solution set.

**Solution**

The answer is false.

Example 3.2.5

True or false: A closed circle on a graph means that the number is included in the solution set.

**Solution**

The answer is true.

### Review

Graph the solution for each inequality on the given number line.

- x<−3

- x>−5

- n≤2

- 1≤n

- x>6

- n<−5

- n≤−6

- x≥−9

- x≤−2

- x>−5

- x≤−8

- x>−10

- x<−6

- x>−7

- x≤−8

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.13.

### Additional Resources

PLIX: Play, Learn, Interact, eXplore: **Number Lines: Freezing Cold Comparison**

Video:

Practice: **Inequalities on a Number Line**

Real World Application: **Zoo Value**