Lesson 8Writing and Graphing Inequalities

Learning Goal

Let’s write inequalities.

Learning Targets

I can graph inequalities on a number line.
I can write an inequality to represent a situation.

Warm Up: Estimate Heights of People

Problem 1

Here is a picture of a man.

A picture of a man standing next to a folding chair. The top of the folding chair comes up to the man's hip.

Name a number, in feet, that is clearly too high for this man’s height.
Name a number, in feet, that is clearly too low for his height.
Make an estimate of his height.

Pause here for a class discussion.

Problem 2

Here is a picture of the same man standing next to a child.

A picture of a man and boy standing next to each other. The top of the boy's head comes up to the chest of the man.

If the man’s actual height is 5 feet 10 inches, what can you say about the height of the child in this picture?

Be prepared to explain your reasoning.

Activity 1: Stories About 9

Problem 1

Drag the green and red open points over the blue points, matching each story, graph, and description.

open applet in presentation mode

Record your results with a number followed by a capital letter and a lowercase letter. For example, 1Aa would mean that responses “1”, “A”, and “a” all represent the same situation.

Print Version

Your teacher will give you a set of paper slips with four stories and questions involving the number 9. Match each question to three representations of the solution: a description or a list, a number line, or an inequality statement.

Problem 2

Compare your matching decisions with another group’s. If there are disagreements, discuss until both groups come to an agreement. Then, record your final matching decisions here.

A fishing boat can hold fewer than 9 people. How many people ( $x$ ) can it hold?
- Description or list:
- Number line:
- Inequality:
Lin needs more than 9 ounces of butter to make cookies for her party. How many ounces of butter ( $x$ ) would be enough?
- Description or list:
- Number line:
- Inequality:
A magician will perform her magic tricks only if there are at least 9 people in the audience. For how many people ( $x$ ) will she perform her magic tricks?
- Description or list:
- Number line:
- Inequality:
A food scale can measure up to 9 kilograms of weight. What weights ( $x$ ) can the scale measure?
- Description or list:
- Number line:
- Inequality:

Activity 2: How High and How Low Can It Be?

Here is a picture of a person and a basketball hoop. Based on the picture, what do you think are reasonable estimates for the maximum and minimum heights of the basketball hoop?

Problem 1

Complete the first blank in each sentence with an estimate, and the second blank with “taller” or “shorter.”
I estimate the minimum height of the basketball hoop to be feet; this means the hoop cannot be than this height.
I estimate the maximum height of the basketball hoop to be feet; this means the hoop cannot be than this height.

Problem 2

Write two inequalities—one to show your estimate for the minimum height of the basketball hoop, and another for the maximum height. Use an inequality symbol and the variable $h$ to represent the unknown height.

Problem 3

Plot each estimate for minimum or maximum value on a number line.

Minimum:

Maximum:

Problem 4

Suppose a classmate estimated the value of $h$ to be 19 feet. Does this estimate agree with your inequality for the maximum height? Does it agree with your inequality for the minimum height? Explain or show how you know.

Problem 5

Ask a partner for an estimate of $h$ . Record the estimate and check if it agrees with your inequalities for maximum and minimum heights.

Are you ready for more?

Problem 1

Find 3 different numbers that $a$ could be if $| a | < 5$ . Plot these points on the number line. Then plot as many other possibilities for $a$ as you can.

Problem 2

Find 3 different numbers that $b$ could be if $| b | > 3$ . Plot these points on the number line. Then plot as many other possibilities for $b$ as you can.

Lesson Summary

An inequality tells us that one value is less than or greater than another value.

Suppose we knew the temperature is less than $3^{\circ} F$ , but we don’t know exactly what it is. To represent what we know about the temperature $t$ in $^{\circ} F$ we can write the inequality: $t < 3$

The temperature can also be graphed on a number line. Any point to the left of 3 is a possible value for $t$ . The open circle at 3 means that $t$ cannot be equal to 3, because the temperature is less than 3.

A number line from -5 to 5. There is an open circle at 3 with an arrow extending to the left from the circle to -5.

Here is another example. Suppose a young traveler has to be at least 16 years old to fly on an airplane without an accompanying adult.

If $a$ represents the age of the traveler, any number greater than 16 is a possible value for $a$ , and 16 itself is also a possible value of $a$ . We can show this on a number line by drawing a closed circle at 16 to show that it meets the requirement (a 16-year-old person can travel alone). From there, we draw a line that points to the right.

A number line from 10 to 28, with tick marks counting by 2's. There is a filled in circle at 16 with an arrow extending to the right from the circle to 28.

We can also write an inequality and equation to show possible values for $a$ : $a > 16$ $a = 16$