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## Introduction

If you have ever been to an amusement park, you are familiar with the height requirements on many of the rides.  As you board the ride, there is a sign stating that riders must be 36, 42, 46, 48, etc. inches tall to ride.  To make it easy for people to determine whether they are eligible to ride, a wooden theme park character is often placed beside the sign.  Anyone who is taller than the character is able to legally get on the ride.

While an equation is a comparison that states two values are equal, an inequality is a comparison between two values that are not equal.  Inequalities differentiate between larger and smaller numbers.

There are four main inequalities that differentiate a larger number and a smaller one.  A fifth type of inequality simply states that two numbers are not equal.

Each inequality pretty much means what it says.  For example, the inequality  >  simply means “greater than.”  In the context of comparing two values, a > b means that the value of a is greater than the value of b.

## Lesson

A simple equation has just one answer.  For the equation x = 3, it is actually possible to put any number in for x.  However, the only number that makes the equation true is the number 3.  The equation x = 3 can be shown on the number line by using a dot.

The inequality x > 3 still has the same point (3) as its anchor on the number line.  However, there are an infinite number of points that can be drawn on the number line that are greater than 3.  Numbers such as 4, 5, 6, and 7 are all greater than three as are decimals like 4.78, 5.2, 8.4, and 11.9745.  It would take an eternity to draw each point individually, so the accepted method of showing all the answers is starting from the number 3 and shading over all the numbers greater than 3.

The graph above has an open circle on the 3 and is shaded on all the numbers to the right of the 3.  The numbers that have been shaded are solutions to x > 3.  The number line appears to end at 8, so the arrow to the right can be shaded to show that the solutions continue indefinitely for numbers larger than 8.

The process of drawing the graph of an inequality can be broken down into three steps.  The three steps for the inequality b ≤ 5 are shown below.

Here are some hints for the three steps above:

Step 1: When drawing the number line, make sure it includes the critical point.  A quick number line can be drawn by putting on just two numbers: the critical number from the problem and the number zero as a reference point.

Step 2: Choosing which circle to use is a matter of asking whether the critical number is a solution or not.  A closed circle means that a solution can equal the critical number, so for ≥ and ≤ a closed circle should be used.  An open circle means that the critical number cannot be a solution, so for > and < an open circle should be used.

Step 3: Many elementary teachers teach that the inequality “eats” the bigger number.  It is common to think of an inequality symbol as a mouth that “eats” the larger number.  Here are some examples:

Thinking of an inequality as “eating” the larger number and it will be easier to decide whether to shade to the right or the left of the critical point.  Take a look at the examples below and guess what the solution would look like.  Then, scroll over the number line and check your answer.

Example 1:

Draw the graph of a ≤ 3

Draw the graph of b > -15

Draw the graph of c < 0

Draw the graph of d ≥ -3

In all of the examples above, it is assumed that solutions include all the fractions and decimals in between each integer.  This is usually the case, but there are problems where fractions and decimals simply don’t make sense.  This is true when considering a real life situation where all logical outcomes are countable (such as ages or a number of items).  Example 2 shows a different type of graph that only includes integers.

Example 2:  All children under the age of 12 eat for free on Thursdays at Gina’s Café.  Draw the graph of the inequality for this situation.

Ages are represented with whole numbers, so the only numbers to consider are whole numbers.  Twelve is not a solution, but all whole numbers less than 12 are solutions.

Drawing the graph using individual circles may not seem as useful as shading, but most math courses include at least a few examples of inequalities involving integers and whole numbers.  If you spot one or more of these problems on an assignment, test, or quiz, remember to place a dot on each individual whole number solution.

## Try It

Draw the graph of each inequality.

1)  r ≤ 5

2)  s > -2

3)  t ≤ 0

4)  u ≥ 44

1)

2)

3)

4)