Solutions of Inequalities
9.1: Unknowns on a Number Line (10 minutes)
The purpose of this warm-up is for students to compare and name values on a number line based on their relative position to one another and 0. Students also review completing inequality statements based on their comparisons.
Since there are many ways to make each inequality true, it may not be possible for students to share all of the possible ways due to time. Consider sharing 2 possibilities for each before moving on to the next question.
Arrange students in groups of 2. Give students 5 minutes of quiet work time. Tell students there are many possible answers for the questions. Give students 1 minute to compare their responses to their partners and decide if they are both correct, even if they are different. Follow with whole-class discussion.
The number line shows several points, each labeled with a letter.
Fill in each blank with a letter so that the inequality statements are true.
- _______ > _______
- _______ < _______
- Jada says that she found three different ways to complete the first question correctly. Do you think this is possible? Explain your reasoning.
- List a possible value for each letter on the number line based on its location.
Ask students to share their responses and explanations for each question. Record and display their responses for all to see. If possible, as students share, record their reasoning directly on the displayed number line and reference the points’ locations.
If there is time after students share the possible values of each point in the last question, ask students how they could complete the inequality: __+__>__. Ask students to share their responses and explanations for how they know the inequality is true based on their assigned values for each point.
9.2: Amusement Park Rides (25 minutes)
The purpose of this activity is for students to represent situations with inequalities and investigate whether values are solutions to multiple inequalities at the same time. Students are formally introduced to the term solution to an inequality and are given the opportunity to use it precisely during discussion (MP6). A solution to an inequality is a value of the variable that makes the inequality true. Students explore these ideas using given height restrictions for a variety of amusement park rides. Students represent the height restrictions as inequality statements and graph those inequalities on the number line. Students reason abstractly when determining whether a value is a solution to one or more of the inequalities and what that means in context (MP2).
Question 3 will likely lead to a discussion of whether or not the endpoints of the inequality are included. As students work, take note that students’ inequalities and graphs should match their reasoning on the inclusion or exclusion of the endpoints 55 and 72. Monitor for one student who included 55 and 72 as a solution and one student who did not.
Prior to beginning this activity, remind students that in their previous work with inequalities, they considered which values made an inequality true and which values did not. Introduce the more formal definition of solution here by using students’ previous work with solutions to equations as a starting point. Just as a solution to an equation was a value of the variable that made the equation true, a solution to an inequality is a value of the variable that makes the inequality true. But while the equations students solved in the last unit generally had one solution, inequalities have many, sometimes infinitely many, solutions.
Arrange students in groups of 2. Give students 5 minutes of quiet think time for questions 1 and 2. Pause after question 2 to tell students to work with their partner for questions 3 through 6. Tell students that if there is disagreement, work to reach agreement. Give students 5 minutes of work time. Follow with a whole-class discussion.
Priya finds these height requirements for some of the rides at an amusement park.
|To ride the . . .||you must be . . .|
|High Bounce||between 55 and 72 inches tall|
|Climb-A-Thon||under 60 inches tall|
|Twirl-O-Coaster||58 inches minimum|
Write an inequality for each of the the three height requirements. Use \(h\) for the unknown height. Then, represent each height requirement on a number line.
Pause here for additional instructions from your teacher.
- Han’s cousin is 55 inches tall. Han doesn’t think she is tall enough to ride the High Bounce, but Kiran believes that she is tall enough. Do you agree with Han or Kiran? Be prepared to explain your reasoning.
Priya can ride the Climb-A-Thon, but she cannot ride the High Bounce or the Twirl-O-Coaster. Which, if any, of the following could be Priya’s height? Be prepared to explain your reasoning.
- 59 inches
- 53 inches
- 56 inches
- Jada is 56 inches tall. Which rides can she go on?
- Kiran is 60 inches tall. Which rides can he go on?
The inequalities \(h < 75\) and \(h > 64\) represent the height restrictions, in inches, of another ride. Write three values that are solutions to both of these inequalities.
Are you ready for more?
Represent the height restrictions for all three rides on a single number line, using a different color for each ride.
- Which part of the number line is shaded with all 3 colors?
- Name one possible height a person could be in order to go on all three rides.
The discussion should include these topics:
- What does it mean to be a solution to an inequality? (A value of the variable that makes the statement true.)
- What are some ways to find solutions to an inequality? (A substitution can be used to check if values make the inequality true, or the inequality can be graphed on a number line and the value checked against the shaded part.)
- How many solutions are there to the inequality for the high bounce? In other words, how many possible heights are there that are allowed? (There are infinitely many solutions because 58.1, 58.01, 58.001, 58.0001, etc. are all allowable heights in inches.)
- Continue the discussion started in the last lesson on the meaning of an open circle vs. a closed circle on the number line. This will come up when students are writing inequalities for the High Bounce ride and then trying to determine if Han’s cousin can go on this ride. There is ambiguity in the language “between x and y,” so both interpretations should be included in the discussion. Take time to make sure students’ inequalities and number lines match their reasoning. Select previously identified students to share their graphs of the height restrictions for the High Bounce ride.
Supports accessibility for: Conceptual processing; Language; Memory
Design Principle(s): Optimize output (for explanation); Cultivate conversation
9.3: What Number Am I? (15 minutes)
This activity is optional due to time considerations. The purpose of this activity is for students to reason about whether given values make an inequality true and justify their answers using inequality statements and graphs (MP3). Students explored this concept in the previous activity, so they should work to articulate how they check if a number is a solution to an inequality statement and an inequality on a number line.
Students will play a game to practice this skill. The game’s goal is for one student to guess a mystery number using as few inequalities as possible. To create a class competition, keep track of how many inequalities each group uses for each number and display scores after each round. The winning group is the group with the lowest score.
Arrange students in groups of 4. As students read the game instructions with their group, give each group 1 set of inequalities and 1 set of numbers, pre-cut from the blackline master. Review the game instructions as a whole class. Students will play 1 round and then pause to reflect and plan strategies for the next rounds. Allow students 10 minutes of game time followed by a whole-class discussion.
Once student groups have completed one round of game play, pause the game and ask groups to reflect on their strategies. Groups should make a plan before continuing to the second round of the game. Post or ask these questions to guide a short discussion:
- “Clue givers, how did you decide which inequalities would be the most helpful for your detective?”
- “Clue givers, did you work together to decide which 3 clues to give or did you decide independently?”
- “Detective, were some inequalities more helpful than others as you tried to guess the mystery number? If so, what made an inequality more helpful?”
Supports accessibility for: Memory; Conceptual processing
Your teacher will give your group two sets of cards—one set shows inequalities and the other shows numbers. Place the inequality cards face up where everyone can see them. Shuffle the number cards and stack them face down.
One person in your group is the detective. The other people will give clues.
Pick one number card from the stack and show it to everyone except the detective.
The people giving clues each choose an inequality that will help the detective identify the unknown number.
The detective studies the inequalities and makes three guesses.
If the detective does not guess the right number, each person chooses another inequality to help.
When the detective does guess the right number, a new person becomes the detective.
Repeat the game until everyone has had a turn being the detective.
Some groups may misunderstand the directions, thinking that each person giving clues is supposed to take a different number card from the stack. Explain to them that there is only one unknown number per round and everyone gives clues about this same number. When a new person becomes the detective, that is when a new number card is drawn.
Once all students have had a turn as the detective, start a whole-class discussion. Ask groups to share their most successful strategies for choosing helpful clue inequalities as well as strategies for using those clues to guess the mystery number. Emphasize how students can check that a value is or is not a solution to an inequality: both symbolic statements (substitute the value and check that the resulting statement is true) and on the number line (plot the value to test and make sure that it falls within a shaded region).
Design Principle(s): Maximize meta-awareness
Ask students to think about situations where a quantity can take on a range of values (for example, the ages of students eligible for a certain program, the salary range that applies to a particular tax rate, the speed you can drive on the highway). Ask students to decide on a variable and represent their situation with two inequality statements. Ask whether the maximum and minimum are included in the range of possible values of the variable (for example, can your maximum speed on a highway be equal to 65 miles per hour or does it have to be less than 65?). Then ask them to graph the solutions on one or two number lines. Invite selected students to share their situations, inequalities, and graphs.
9.4: Cool-down - Solutions of Inequalities (5 minutes)
Student Lesson Summary
Let’s say a movie ticket costs less than \$10. If \(c\) represents the cost of a movie ticket, we can use \(c < 10\) to express what we know about the cost of a ticket.
Any value of \(c\) that makes the inequality true is called a solution to the inequality.
For example, 5 is a solution to the inequality \(c < 10\) because \(5<10\) (or “5 is less than 10”) is a true statement, but 12 is not a solution because \(12<10\) (“12 is less than 10”) is not a true statement.
If a situation involves more than one boundary or limit, we will need more than one inequality to express it.
For example, if we knew that it rained for more than 10 minutes but less than 30 minutes, we can describe the number of minutes that it rained (\(r\)) with the following inequalities and number lines. \(\displaystyle r > 10\)
Any number of minutes greater than 10 is a solution to \(r>10\), and any number less than 30 is a solution to \(r<30\). But to meet the condition of “more than 10 but less than 30,” the solutions are limited to the numbers between 10 and 30 minutes, not including 10 and 30.
We can show the solutions visually by graphing the two inequalities on one number line.