158. Understanding and Representing Inequalities

Learning Intentions

• To understand that an inequality is a mathematical statement that one value is larger than (or as large as) another value
• represent inequalities on a number line using open or closed circles and/or arrows
• describe real-life situations using inequalities

Pre-requisite Summary

• An equation states that two expressions are equal, but an inequality compares values that may not be equal
• The symbols $>$, $<$, $\ge$ and $\le$ compare numbers
• A number line can be used to show the position of numbers relative to each other
• An open circle shows that an endpoint is not included in the solution set
• A closed circle shows that an endpoint is included in the solution set
• An arrow on a number line shows that the inequality continues indefinitely in one direction
• Real situations often involve limits, minimums and maximums, which can be described using inequalities

Worked Examples

Worked Example 1

State in words what each inequality means.
a) $x>4$
b) $y\le 7$

Worked Example 2

Represent each inequality on a number line.
a) $x<3$
b) $x\ge -2$

Worked Example 3

Describe a real-life situation that could be represented by $a\ge 18$.

Worked Example 4

Write an inequality for the statement: a ticket costs no more than $12.

Worked Example 5

Represent $m\le 5$ on a number line and explain why a closed circle is used.

Worked Example 6

Describe a real-life situation that could be represented by $t<10$.

Problems

Problem 1

State in words what each inequality means.
a) $x<6$
b) $p\ge 2$

Problem 2

Represent each inequality on a number line.
a) $x>1$
b) $x\le -4$

Problem 3

Describe a real-life situation that could be represented by $n\le 30$.

Problem 4

Write an inequality for the statement: a bag must weigh at least $5$ kg.

Problem 5

Represent $k>0$ on a number line and explain why an open circle is used.

Problem 6

Describe a real-life situation that could be represented by $h\ge 150$.

Exercises

Understanding and Fluency

1. Write each inequality in words.
   a) $x>5$
   b) $y<9$
   c) $a\ge 4$

2. Write each inequality in words.
   a) $m\le 12$
   b) $p>-3$
   c) $q\ge 0$

3. For each inequality, state whether the endpoint is included.
   a) $x>2$
   b) $x\le 7$
   c) $x\ge -1$

4. Represent each inequality on a number line.
   a) $x<4$
   b) $x\ge 6$
   c) $x\le -2$

5. Represent each inequality on a number line.
   a) $x>-5$
   b) $x\le 3$
   c) $x\ge 1$

6. Match each inequality to the correct circle type.
   a) $x<8$
   b) $x\ge 8$
   c) $x\le 8$

7. Write an inequality for each description.
   a) a number is greater than $11$
   b) a number is at most $6$
   c) a number is at least $2$

8. Write an inequality for each description.
   a) a temperature is below $0^{\circ }\text{C}$
   b) a speed is no more than $50$ km/h
   c) a person is at least $16$ years old

Reasoning

9. Explain the difference between $x>3$ and $x\ge 3$.

10. A student uses a closed circle to represent $x<5$. Explain the error.

11. Explain why an arrow is needed when representing an inequality such as $x\le 2$ on a number line.

12. Noah says that $p\le 7$ means $p$ is smaller than $7$ but not equal to $7$. Explain why Noah is incorrect.

Problem-solving

13. A ride at a theme park can be used only by people whose height $h$ is at least $140$ cm. Write an inequality for this situation.

14. A cinema gives a child ticket to anyone younger than $15$. Write an inequality for the person's age $a$.

15. A suitcase must weigh no more than $20$ kg at check-in. Write an inequality for the suitcase mass $m$.

16. A school hall can hold at most $300$ people. Write an inequality for the number of people $n$ in the hall.

17. The outside temperature is forecast to stay above ${18}^{\circ }\text{C}$ all day. Write an inequality for the temperature $T$.

18. A water tank must contain at least $50$ L before the pump can be turned on. Write an inequality for the amount of water $w$ in the tank.

Potential Misunderstandings

• Thinking an inequality is the same as an equation
• Confusing the meaning of $>$ and $<$
• Forgetting that $\ge$ and $\le$ include equality
• Using an open circle when the endpoint should be included
• Using a closed circle when the endpoint should not be included
• Drawing the arrow in the wrong direction on the number line
• Thinking an inequality has only one solution rather than many possible values
• Misreading phrases such as “at least”, “at most”, “no more than” and “less than”
• Writing an equation instead of an inequality for a real-life situation
• Forgetting to define what the variable represents in context