<p>Students use what they learned in the previous two lessons to solve a system of inequalities and apply their new understanding to real-world linear programming-type problems. Students will:<br>- graph linear inequalities and highlight the appropriate portion of the solution set.<br>- use technology to create graphs of inequality systems.<br>- evaluate the characteristics of figures formed by graphing inequalities.</p>

<p>- How would you characterize the relationship between quantities represented by linear equations or inequalities?&nbsp;<br>- How would you solve a system of equations using graphical and/or algebraic techniques, and how would you interpret the results?&nbsp;<br>- How can we demonstrate that algebraic properties and processes are extensions of arithmetic properties and processes, and how can we apply algebraic properties and processes to problems?&nbsp;<br>- Which functional representation would you use to model a real-world situation, and how would you explain your solution to the problem?</p>

<p><span style="background-color:rgb(255,255,255);color:rgb(8,42,61);"><i>(Vocabulary hasn't been entered into the lesson plan.)</i></span></p>

<p>- graph paper&nbsp;<br>- rulers&nbsp;<br>- copies of Graphing Inequalities Worksheet (A1-5-3_Graphing Inequalities Worksheet)</p>

<p>- Teacher observation during classroom discussion and lesson activities&nbsp;<br>- Random Reporter&nbsp;<br>- Partner Problem</p>

<p><strong>T:</strong> Use the following ways to adjust the lesson to the requirements of your students throughout the year.&nbsp;<br><br><strong>Routine:</strong> Partner and group work should be implemented throughout the lesson to facilitate mutual assistance among students. The emphasis should be on communicating mathematical ideas using vocabulary phrases&nbsp;appropriate for the topic. Accurate notes, as well as active participation, are required for the class. Assist students with organized and note-taking abilities to improve their learning experience and create a useful resource (notes).&nbsp;<br><br><strong>Partner or Small-Group Activity: </strong>Use the same small-group problem (or an equivalent problem) from Lesson 2, but modify it to include inequalities.&nbsp;<br><br>Solve the system of inequalities, create the graphs, and find the region that determines the solution.&nbsp;</p><figure class="image"><img style="aspect-ratio:101/49;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_20.png" width="101" height="49"></figure><p><strong>Extension:</strong> Have students work in pairs to solve and graph three or four inequalities. Shade the region to identify the solution. Students should solve the problem by hand, then on graph paper, and finally with technology, such as a graphing calculator, if available. Request that students create a presentation with a visual display, such as a poster, PowerPoint, transparency, etc. If time allows, students can:&nbsp;<br><br>· Exchange problems with another group and evaluate their responses.&nbsp;<br><br>· Utilize the problem to instruct the entire class or a small group.&nbsp;<br><br>· Present their answers (with visuals) and explain the most efficient technique.&nbsp;<br><br><strong>O:</strong> This lesson begins with a discussion of a problem from the previous lesson, which is then connected to new content by altering the scenario to inequalities. Students develop an understanding of the relationship between equation systems and inequalities. They are exposed to the concept of linear programming and the use of inequalities to calculate the area or perimeter of geometric forms. This class concludes with a linear programming challenge that students can apply to their&nbsp;situations.&nbsp;</p>

<p><strong>W:</strong> This lesson relates the student's previous knowledge of systems of equations to the new concept of systems of inequalities. Although students realize that solving systems of equations is typically more efficient when done algebraically, they will discover that solving systems of inequalities by graphing is the most common (and easiest) conceptual technique. Students will use linear programming to solve systems of two inequalities and then graph problems of three or more inequalities. Students will measure the attributes of the geometric shapes formed by graphing these inequalities.&nbsp;<br><br><strong>H:</strong> <strong>"Remember the problem with movie ticket prices from the previous lesson? The local cinema was celebrating its 25th anniversary and offering a discount on tickets. You had no idea what the ticket costs were, but you heard two of your classmates discussing how much it cost their families to go to the movies last weekend. One student reported that the cost for two adults and two children was $12. Another student said that the fee for three adults and four children was $20."</strong><br><br><strong>"What would the equations look like if you heard your first classmate say she couldn't remember the exact total, but it was $12 or less for two adults and two children? The second student couldn't remember the exact total either, but it was $20 or less for three adults and four children."</strong><br><br><strong>"How would the system appear now? Has any of the data changed?"&nbsp;</strong><br><br><strong>If the numbers remained constant, how did the equation change?"&nbsp;</strong><br><br>Allow students to work together to solve the problem. Inform them that students will be selected at random from all groups to present their solutions to the class (using the Random Reporter approach).<br><br>When groups have completed their presentations, they should take notes on the lesson. Hand out a copy of the notes to students who will be distracted by taking notes, allowing them to focus more on the teacher's spoken descriptions and instructions. &nbsp;<br><br><strong>E: </strong>Encourage students to practice graphing inequalities (M-A1-5-3_Graphing Inequalities Worksheet).&nbsp;<br><br>The following notes should be posted on the board for students to copy:&nbsp; &nbsp;<br><br><strong>Graphing Systems of Linear Inequalities</strong><br><br><u>Example 1:</u>&nbsp;</p><figure class="image"><img style="aspect-ratio:93/75;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_21.png" width="93" height="75"></figure><p>In the Graphing Inequalities Worksheet, highway signs tell us to travel <strong>as</strong> fast <strong>or</strong> faster than 40 miles per hour and <strong>no</strong> faster than 65 miles per hour. The number line 40–65 represents the inequality in one dimension (a straight line) and two directions (left and right). Similarities, the system's two inequalities have two dimensions (<i>x</i>-axis and <i>y</i>-axis) and four directions (left, right, up, and down).<br><br>(End of student notes.)<br><br>Step 1: Convert the inequality expressions into slope-intercept form if they aren't already.<br><br>The first equation is already in slope-intercept form, however,&nbsp;the second equation must be changed.<br><br>Write the following notes on the board for students to copy.<br><br>***Remember that if you divide both sides by a negative value, the inequality sign changes.<br><br>Step 2: Graph inequalities as equations.<br><br>a. Graph a solid line if the inequality is ≥ or ≤ to include every point along the line.<br><br>b. If the inequality is &gt; or &lt;, create a dashed line to serve as a border without including the points directly on the line.<br><br>In this example, one line is solid (&gt;), while another is broken (≤). The drawn lines represent the borders for <i>solution set</i>.</p><figure class="image"><img style="aspect-ratio:337/333;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_22.png" width="337" height="333"></figure><p>Step 3: Shade the solution set. Because this is a linear inequalities system, there will be more than one answer. All coordinates that fulfill the system of inequalities are part of the <i>solution set</i>. Select a test point (not on either of the lines) to select which side of the lines to shade.<br><br>a. Fill in the shading around the test point if it satisfies both inequalities.<br><br>b. If the test point does not meet both inequalities, choose a different place where the lines intersect and select another test point.<br><br>In this example, select the test point (4, 2).<br><br>Is 2&nbsp;≤&nbsp;2(4) + 1? Is 2 ≤ 9? Yes</p><p>Is 4 &gt; -\(1 \over 3\)(2) + 3? Is 4 &gt; \(1 {1 \over 3} \)? Yes<br><br>Shade the bordered area that contains the point (4, 2) because the test point satisfies both inequalities.<br><br>(End of student notes.)<br><br><br><u>Example 2:</u>&nbsp;</p><figure class="image"><img style="aspect-ratio:90/68;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_23.png" width="90" height="68"></figure><p><strong>Note: This solution must be shown graphically.&nbsp;</strong><br><br>Step 1: Convert the inequality expressions into slope-intercept form if they aren't already.<br><br>The first equation is already in slope-intercept form, but the second one needs to be modified.</p><figure class="image"><img style="aspect-ratio:103/95;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_24.png" width="103" height="95"></figure><p>Note that the inequality sign was reversed..<br><br><br><strong>"Can anyone explain why?"</strong><i>(because both sides were divided by a negative number)</i><br><br>Step 2: Draw the inequality graphs like equations.<br><br>a. Draw a solid line if the inequality is ≥ or ≤. <strong>"Who knows why?"</strong><br><br>b. Draw a dashed line on the graph if the inequality is &gt; or &lt;.<strong> "Who knows why?"</strong><br><br>In this example, the first line will be dashed while the second will be solid.<br><br>Step 3: Make the solution set shaded.<br><br>a. Fill in the shading around the test point if it satisfies both inequalities.<br><br>b. Choose a different area that the lines border and choose a different test point if the current test point does not satisfy both inequalities.</p><figure class="image"><img style="aspect-ratio:379/311;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_25.png" width="379" height="311"></figure><p><br>In this example, pick the test point (0, 0).<br><br>Is 0 &lt; –\(2 \over 3\)(0) + 4? Is 0 &lt; 4? Yes<br><br>Is 2(0) – 3(0) ≥ 9? Is 0 ≥ 9? No<br><br>A new test point must be chosen because the current one only meets the first inequality. Select a point that is opposite the previous test point's solid line but on the same side of the dashed line. Try (6, -1).<br><br>Ask students, <strong>"Has anyone noticed a faster way to know which side of the line to shade?"</strong> if they haven't seen that the graphs of &gt; are always shaded above the line and &lt; below the line. Students can have trouble distinguishing between what is "above" and "below," particularly when the line has a steep slope.<br><br><strong>E:</strong> Assign the students to solve the four systems listed below. They are able to work alone or in pair. &nbsp;</p><figure class="image"><img style="aspect-ratio:281/122;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_26.png" width="281" height="122"></figure><p>Discuss the solutions as a whole class.<br><br><strong>"The first problem (the cost of cinema tickets) introduces us to the idea of linear programming. A technique known as linear programming can be used to find the optimal result given a set of restrictions, such as 'maximizing profits' or 'minimizing costs'."</strong><br><br><strong>“What changed in the problem after students stated that the movie cost exactly $12 and $20 to the movie cost $12 or less and $20 or less? The equations turned into inequalities.&nbsp;”</strong></p><figure class="image"><img style="aspect-ratio:100/56;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_27.png" width="100" height="56"></figure><p><strong>“Now that we have these inequality graphs, let's keep in mind that </strong><i><strong>x</strong></i><strong> and </strong><i><strong>y</strong></i><strong> stand for prices. This indicates that the system needs to have two more constraints added to it. The two inequalities that follow—</strong><i><strong>x</strong></i><strong> ≥ 0 and </strong><i><strong>y</strong></i><strong> ≥ 0—will be included because movie theaters cannot sell tickets for less than $0.”</strong></p><figure class="image"><img style="aspect-ratio:98/107;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_28.png" width="98" height="107"></figure><p>Ask students to graph this linear programming problem and then analyze the implications of the solution set for the cost of movie tickets.<br><br><strong>"Measuring attributes of the geometric shapes created by the borders of the solution set is another reason we use systems of inequalities."</strong><br><br><u>Example:</u> According to the following system of inequalities, the surface of a desk can be described as:</p><figure class="image"><img style="aspect-ratio:57/98;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_29.png" width="57" height="98"></figure><p>Calculate the area and perimeter of the desk by graphing the system mentioned above.<br><br>Ask students to work in pairs to find the area of the form that results from graphing the following system of inequalities.</p><figure class="image"><img style="aspect-ratio:90/79;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_30.png" width="90" height="79"></figure><p>Review the answer as a class.<br><br><strong>R:</strong> Assign students to graph the following systems of inequalities in groups of three, and have them shade the appropriate solution set. Every student graphs a single border line, and in order to identify the shaded area, students must select test points that are distinct from one another.</p><figure class="image"><img style="aspect-ratio:379/72;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_31.png" width="379" height="72"></figure><p>After everyone has finished, have groups merge to form a larger group of six. Students should discuss and compare their work. Next, talk about the graphs as a class and whether there were any differences between the groups. Ask students to turn in their completed projects.<br><br><strong>E:</strong> <strong><u>Partner Problem:</u></strong><br><br>Introduce the following problem to the students. Give them a hint that they should probably figure out what the <i>x</i> and <i>y</i>-variables will represent first. Before letting them proceed to finish the other steps, assist them if they are having trouble with this portion of the problem.<br><br>Charlie has two jobs. He works 10 hours or fewer per week at the school's concession stand during home athletic games. In addition, he works 20 hours or fewer a week at the nearby fast-food business. At the fast food restaurant, he makes $8 per hour, but at the concession stand, he makes $6. Charlie wants to make more money each week than $56.<br><br>1. Create the inequality system that best describes Charlie's situation.<br><br>2. Draw a graph of the inequality system and shade the solution set.<br><br>3. What does the shaded area represent in terms of the hours Charlie works and the amount of money he earns?</p>