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<h4>Activity (20 minutes)</h4>
<p>This activity encourages students to interpret an inequality and its solution set in terms of a situation. A context can help students intuit why the solutions to an inequality form a ray on the number line.</p>
<p>The activity also prompts students to think about the solutions to an inequality in terms of a related equation. Here, the situation involves choosing between two options. An equation can be written to represent the two options being equal. The solution to that equation can be seen as a tipping point, on either side of which one option would be better.</p>
<p>Students may recall this way of solving inequalities from middle school, but they may also solve by testing different possible values or by reasoning about the relationship between quantities in other ways. The work here encourages students to reason quantitatively and abstractly and to make sense of problems and persevere in solving them.</p>
<p>Monitor the strategies students use to find the solutions to \( 9(n+3)<10(n+1) \), and identify students using different approaches. Students may:</p>
<ul>
<li>Try different values of \(n\) until the inequality is no longer true.</li>
<li>Try different numbers higher than 12 (based on their work on the first question) and find that, up to 17 students, the cost to go to Orchard B is lower. Beyond 17 students, the cost for Orchard A is lower.</li>
<li>Solve the equation \( 9(n+3)=10(n+1) \) to find the number of students at which the costs for both options would be equal. That number is 17. Then, try a higher or lower number to see which side of the equation has a smaller value.</li>
<li>Reason about the difference in the cost per student and cost for chaperones. The cost per student at Orchard A ($9) is $1 lower than at Orchard B ($10). But because 3 chaperones are required at Orchard A ($27 for 3 chaperones) and only 1 at Orchard B ($10 for 1 chaperone), the cost for chaperones is $17 higher at Orchard A than at Orchard B. So if 17 students go on the trip, the cost would be the same at both places. If more than 17 students go, Orchard A would be cheaper.</li>
</ul>
<p>Some students may find the solutions to \( 9(n+3)<10(n+1) \) by manipulating the inequality to isolate \(n\). Depending on the operations performed, they may once again end up with an incorrect solution set if they forget to reverse the inequality symbol. (For example, in the final step of solving, they may go from \( -1n<-17 \) to \( n<17 \).) If this happens, bring the issue to students’ attention during activity synthesis.</p>
<h4>Launch</h4>
<p>Read the first part of the task statement with the class and make sure students understand the given information.<br>
</p>
<p>Arrange students in groups of 2. For the first set of questions, ask one partner to find the cost of going to Orchard A and the other partner to find the cost of going to Orchard B, and then compare the costs. Before students move on to the second set of questions, pause to hear which option works best for 8, 12, and 30 students.</p>
<h4>Student Activity</h4>
<p>For questions 1–3, use the information about the orchards to answer each question. </p>
<blockquote>
<p>A teacher is choosing between two options for a class field trip to an orchard. At each orchard, the same price applies to both chaperones and students.</p>
<ul>
<li>At Orchard A, admission costs $9 per person, and 3 chaperones are required.</li>
<li>At Orchard B, the cost is $10 per person, but only 1 chaperone is required.</li>
</ul>
<img alt class="img-fluid atto_image_button_text-bottom" height="192" role="presentation" src="https://k12.openstax.org/contents/raise/resources/32fca147f954f8245a5eb3ffa647f50ac4b841c5" width="300"> <br>
<br>
</blockquote>
<ol class="os-raise-noindent">
<li>Which orchard would be cheaper to visit if the class has 8 students?
<ul>
<li>Orchard A</li>
<li>Orchard B</li>
</ul>
</li>
</ol>
<p><strong>Answer: </strong>Orchard B</p>
<ol start="2">
<li>Which orchard would be cheaper to visit if the class has 12 students?
<ul>
<li>Orchard A</li>
<li>Orchard B</li>
</ul>
</li>
</ol>
<p><strong>Answer: </strong>Orchard B</p>
<ol start="3">
<li>Which orchard would be cheaper to visit if the class has 30 students?
<ul>
<li>Orchard A</li>
<li>Orchard B</li>
</ul>
</li>
</ol>
<p><strong>Answer: </strong>Orchard A</p>
<p>Use the following information to answer questions 4–7.</p>
<blockquote>
<p>To help her compare the cost of her two options, the teacher first writes the equation \( 9(n + 3) = 10(n + 1) \), and then she writes the inequality \( 9(n + 3) < 10(n + 1) \). <br>
</p>
</blockquote>
<ol start="4">
<li>What does \(n\) represent in each statement? </li>
</ol>
<p><strong>Answer: </strong>\( n \) represents the number of students on the field trip.</p>
<ol start="5">
<li>In this situation, what does the equation \( 9(n + 3) = 10(n + 1) \) mean? </li>
</ol>
<p><strong>Answer: </strong>\( 9(n + 3) \) is the cost of going to Orchard A, and \( 10(n + 1) \) is the cost of going to Orchard B. The equation represents the two options costing the same amount.</p>
<ol start="6">
<li>What does the solution to the inequality \( 9(n + 3) < 10(n + 1) \) tell us? </li>
</ol>
<p><strong>Answer: </strong>The solution represents the number of students on the field trip at which it would be cheaper to visit Orchard A.</p>
<ol start="7">
<li>The teacher needs a visual aid to show the school budgeting committee. Graph the solution set to the inequality on the number line. Be prepared to show or explain your reasoning. </li>
</ol>
<p><strong>Answer: </strong>\( n>17 \)</p>
<p><img alt="Inequality on a number line." src="https://k12.openstax.org/contents/raise/resources/b2b58e0109598a3693684702500edcb4f5b38b19"> </p>
<br>
<br>
<h4>Video: Understanding the Meaning of an Inequality </h4>
<p>Watch the following video to learn more about the meaning of an inequality.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container">
<video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/bacd60c30243cf20bdfe33fa8397e80c0dfa8399">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/b381ef0a8fbf9b02917609c33dc6a17686bc820f" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/bacd60c30243cf20bdfe33fa8397e80c0dfa8399 </video>
</div>
</div>
<br>
<br>
<h4>Anticipated Misconceptions</h4>
<p>If students struggle to interpret the meaning of the equation \( 9(n+3)=10(n+1) \) and of the inequality \( 9(n+3)<10(n+1) \), ask them to think about what each side of the equal sign or the inequality symbol represents.<br>
</p>
<h4>Activity Synthesis</h4>
<p>Make sure students understand the meaning of the inequality in context and recognize that there are various ways to find the solutions.</p>
<p>Select previously identified students to share how they found the solution set, in the sequence shown in the Activity Narrative (starting with guessing and checking, and ending with reasoning more structurally). It is not necessary to discuss all the listed strategies, but if the idea of solving a related equation doesn’t come up, point it out.<br>
</p>
<p>Explain that one way to think about the solutions to the inequality is by thinking about the solution to a related equation. In this context, the solution to \( 9(n+3)=10(n+1) \) gives us the number of students at which it would cost the same to go to either orchard. This is a boundary value for \(n\). On one side of the boundary, the cost of Option A would be higher. On the other, it would be lower. We can test a value that is higher and one that is lower than this boundary value to see which one makes the inequality \( 9(n+3)<10(n+1) \) true.</p>
<p>If a student brings up “flipping the symbol when multiplying or dividing by a negative number” as a strategy, invite them to explain why it works. Emphasize that in general it is more helpful and reliable to use reasoning strategies that we understand and can explain. If we use a rule without some idea of how it came about or why it works, we might end up misapplying it (for example, flipping the inequality symbol anytime we see a negative sign, even if we’re simply adding or subtracting). If we forget or misremember the rule, we would be stuck or make errors.</p>
<br>
<!--BEGIN ELL AND SWD GRAY BOX -->
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 5 Co-Craft Questions: Reading, Writing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use this routine to help students consider the context of this problem and to increase awareness of the language of mathematical comparisons. Ask students to keep their books or devices closed and display only the image and the task statement, without revealing the questions that follow. Give students 1–2 minutes to write their own mathematical questions about the situation, and then invite them to share their questions with a partner. Listen for and amplify any questions involving comparing costs. Once students compare their questions, reveal the remainder of the task.</p>
<p class="os-raise-text-italicize">Design Principles: Maximize Meta-awareness, Support Sense-making</p>
<p class="os-raise-extrasupport-title">Learn more about this routine</p>
<p> <a href="https://www.youtube.com/watch?v=P_NQJdG92iA;&rel=0" target="_blank">View the instructional video</a> and <a href="https://k12.openstax.org/contents/raise/resources/4e340aa86ff7eda8a1076cbe2ff84123e50e8012" target="_blank">follow along with the materials</a> to assist you with learning this routine. </p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p> <a href="https://k12.openstax.org/contents/raise/resources/77a07fd176bcc1a05392967ab523ab95586bfc98" target="_blank">Distribute graphic organizers</a> to the students to assist them with participating in this routine. </p>
</div>
</div>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Representation: Internalize Comprehension</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Represent the same information through different modalities by using a table. If students are unsure where to begin, suggest that they use a table to help organize the information provided. Guide students in making decisions about what inputs to include in their table. Be sure that they include 17 and some values directly above and below to support their analysis.</p>
<p class="os-raise-text-italicize">Supports accessibility for: Conceptual processing; Visual-spatial processing</p>
</div>
</div>
<!--END ELL AND SWD GRAY BOX -->
<br>
<h3>2.9.3 Self Check</h3>
<strong><em>Following the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em><br>
</strong><br>
<!--SELF CHECK QUESTION GOES BEFORE THE Table -->
<p class="os-raise-text-bold">QUESTION:</p>
<p>A water park charges $17 per entry plus a $40 fee for raft rentals. An amusement park charges $21 per entry.</p>
<p>For the number of entries, \(e\), which inequality represents the total cost of the water park being less than the total cost of the amusement park?</p>
<!--SELF CHECK table-->
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td><p>\( 17e – 40 < 21e \)</p>
<br></td>
<td><p>Incorrect. Let’s try again in a different way: Should the fee be added or subtracted? The answer is \( 17e + 40 < 21e \).</p></td>
</tr>
<tr>
<td><p>\( 17e + 40 < 21e \)</p></td>
<td><p>That’s correct! Check yourself: The water park is represented by \( 17e + 40 \) and the amusement park by \(21e\). In this inequality, the water park is less than the total cost of the amusement park.</p></td>
</tr>
<tr>
<td><p>\( 17e < 21e + 40 \)</p></td>
<td><p>Incorrect. Let’s try again in a different way: The $40 fee applies to the water park, not the amusement park. The answer is \( 17e + 40 < 21e \).</p></td>
</tr>
<tr>
<td><p>\( 17e + 40 > 21e \)</p></td>
<td><p>Incorrect. Let’s try again in a different way: This represents the total cost of the water park being greater than the total cost of the amusement park. The answer is \( 17e + 40 < 21e \).</p></td>
</tr>
</tbody>
</table>
<br>
<h3>2.9.3: Additional Resources </h3>
<p><strong><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it. </em></strong></p>
<h4>Translating Situations into Inequalities</h4>
<p>Many real-life situations require us to use inequalities. Translating the details of these situations into inequalities is the first step in understanding them. Let’s look at how we can translate these situations into mathematical sentences.</p>
<br>
<p><strong>Example 1</strong> </p>
<p>Imani won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. She can only spend up to the amount of the mini-grant.</p>
<p>Choose a variable to represent the quantity of tablet computers:</p>
<p>Let \( n \) = the number of tablets.</p>
<p>Translate: Write a sentence that gives the information provided.</p>
<p>$254.12 times the number of tablets is no more than $4,000.</p>
<p>Translate into an inequality:</p>
<p>\( 254.12n \leq 4000 \)</p>
<p>Imani’s situation is represented by the inequality \(254.12n \leq 4000\).</p>
<p><strong>Example 2 </strong></p>
<p>Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. Her monthly bill can be no more than $50.</p>
<p>Choose a variable to represent the quantity of text messages:</p>
<p>Let \( t \) = the number of text messages.</p>
<p>Translate: Write a sentence that gives the information provided.</p>
<p>$28.80 plus $0.20 times the number of text messages is less than or equal to $50.</p>
<p>Translate into an inequality:</p>
<p>\( 28.80 + 0.20t \leq 50 \)</p>
<p>Taleisha’s situation is represented by the inequality \( 28.80 + 0.20t \leq 50 \).</p>
<h4>Try It: Translating Situations into Inequalities</h4>
<p>Read each problem below and write an inequality to represent the situation.</p>
<ol class="os-raise-noindent">
<li>Angie has at most $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. She will also buy one bag of chips for $3.99. <br>
<p><strong>Answer: </strong>\( 2.63j + 3.99 ≤ 20 \)</p>
</li>
<li>Jose wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. He must also pay $5 for parking. His budget for the party is less than $500. <br>
<p><strong>Answer: </strong>\( 42.75p + 5 < 500 \)</p>
</li>
</ol>