Lesson 1

Writing and Graphing Inequalities

Let’s write inequalities.

1.1: Estimate Heights of People

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

    Pause here for a class discussion.

  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.

1.2: Stories about 9

  1. Drag the green and red open points over the blue points. Match each story with a numberline and a description.
  2. Think about how you would represent each set of numbers with an inequality statement, talk about it with your partner, and be prepared to discuss your ideas with the class. Then, record your final matching decisions here.  
    1. A fishing boat can hold fewer than 9 people. How many people (\(x\)) can it hold?

      • Description or list:

      • Number line:
        A number line with the numbers 0 through 15 are indicated.

      • Inequality:

    2. 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:
        A number line with the numbers 0 through 15 are indicated.

      • Inequality:

    3. 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:
        A number line with the numbers 0 through 15 are indicated.

      • Inequality:

    4. A food scale can measure up to 9 kilograms of weight. What weights (\(x\)) can the scale measure?

      • Description or list:

      • Number line:
        A number line with the numbers 0 through 15 are indicated.

      • Inequality:

1.3: 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?

  1. Complete the first blank in each sentence with an estimate, and the second blank with “taller” or “shorter.”

    A picture of a basketball hoop, woman, and step ladder. 
    1. I estimate the minimum height of the basketball hoop to be _________ feet; this means the hoop cannot be _____________ than this height.
    2. I estimate the maximum height of the basketball hoop to be _________ feet; this means the hoop cannot be _____________ than this height.

  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.

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

    • Minimum:
      A blank number line with 16 evenly spaced tick marks.
    • Maximum:
      A blank number line with 16 evenly spaced tick marks.
  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.
  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.


  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.
    A number line from negative 10 to 10 by ones.
  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.
    A number line from negative 10 to 10 by ones.

Summary

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

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

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. 

Number line, negative 5 to 5 by ones.  Arrow with open circle begins at 3 and continues left.

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.

Number line, 10 to 28 by twos. Arrow with closed circle begins at 16 and continues to the right.

We can also write an inequality and equation to show possible values for \(a\): \(\displaystyle a>16\) \(\displaystyle a=16\)