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<h4>Activity</h4>
<p>Noah is having trouble solving two equations. In each case, he took steps that he thought were acceptable but ended
up with statements that are clearly not true.</p>
<p>Analyze Noah’s work on each equation and the moves he made. Were they acceptable moves? Why do you think he
ended up with a false equation?</p>
<p>Discuss your observations with your group and be prepared to share your conclusions. If you get stuck, consider
solving each equation.</p>
<div class="os-raise-d-flex os-raise-justify-content-between">
<p><strong>Example 1</strong><br>
<br>
<strong>Step 1 - </strong>Original equation.<br>
\( x+6 =4x+1-3x \)<br>
<br>
<strong>Step 2 - </strong>Apply the Commutative Property.<br>
\( x+6 =4x-3x+1 \)<br>
<br>
<strong>Step 3 - </strong>Combine like terms.<br>
\( x+6 =x+1 \)<br>
<br>
<strong>Step 4 - </strong>Subtract \(x\) from each side.<br>
\( 6 =1 \)
</p>
<p><strong>Example 2</strong><br>
<br>
<strong>Step 1 - </strong>Original equation.<br>
\( 2(5+x)-1=3x+9 \)<br>
<br>
<strong>Step 2 - </strong>Apply the Distributive Property.<br>
\( 10+2x-1 =3x+9 \)<br>
<br>
<strong>Step 3 - </strong>Subtract 10 from each side.<br>
\( 2x-1 =3x-1 \)<br>
<br>
<strong>Step 4 - </strong>Add 1 to each side.<br>
\( 2x=3x \)<br>
</p>
</div>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="43b7e54f-2daf-455d-a614-7205300110fd" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Why did Noah have no solution for either equation? </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> Your answer may vary, but here is an example: <br>
All the moves that Noah made were acceptable, but because he ended up with a false statement, that means there is no value of \(x\) that makes the first equation true. We can see it in the third line: \(x+6=x+1\). There is no number that can make the equation true.</p>
</div>
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ae6510ed-3ed1-4065-b2bd-7d3054b87b40" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2">
<li>Why did Noah arrive at no solution for the equation solved in Example 2?</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> Your answer may vary, but here is a sample. </p><br>
<p>Example 1: All the moves that Noah made were acceptable, but because he ended up with a false statement, that
means there is no value of \( x \) that makes the first equation true. We can see it in the third line: \( x + 6 =
x + 1 \). There is no number that can make the equation true.</p><br>
<p>Example 2: Noah's moves seem acceptable except for the last one. If he had subtracted \( 2x \) from each side of
the equation, he would have \( 0 = x \). Instead, he divided both sides by \( x \) which would give an undefined
number if \( x\;\;is\;\;0 \). </p>
<p>Noah tried to solve a third equation and ended up with a true statement, without variables.</p>
<p><strong>Step 1 - </strong>Original equation.<br></p>
<p>\( 2 (4 + x) - 2 = 2x + 6 \)</p>
<p><strong>Step 2 - </strong>Apply the Distributive Property.<br></p>
<p>\( 8+ 2x - 2 =2x + 6 \)</p>
<p><strong>Step 3 - </strong>Combine Like Terms.<br></p>
<p>\( 2x +6 = 2x +6 \)</p>
<p><strong>Step 4 - </strong>Subtract 6 from both sides. <br></p>
<p>\( 2x = 2x \)</p>
<p><strong>Step 5 - </strong>Subtract \(2x\) from both sides.<br></p>
<p>\( 0=0 \) </p>
<br>
<p>Since there is not a variable and the statement is true, any value of \(x\) would be a solution for the original equation. So, this equation has infinitely many solutions.</p><br>
</div>
</div>
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="eventShowMore1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
</div>
<div class="os-raise-ib-cta-prompt">
</div>
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="dac430db-9805-4437-ac7c-fa4a53c354a7" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>1. We cannot divide the number 100 by zero because dividing by zero is undefined. Instead, try dividing 100 by 10, then 1, then 0.1, then 0.01. What do you notice happens as you divide by smaller numbers? </p>
</div>
<div class="os-raise-ib-input-prompt">
<!-- INSERT ANY VALID HTML HERE -->
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: Your answer may vary, but here is a sample.</p>
<p>10, 100, 1000, 10,000. For example: The quotients are getting very large.</p>
</div>
</div>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="bd7e13ac-f041-4c68-b0e4-a1e67cacb312" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<div class="os-raise-ib-input-content">
<p>2. Now try dividing the number -100 by 10, by 1, by 0.1, 0.01. What is the same and what is different when you compare to question 1?<br></p>
</div>
<div class="os-raise-ib-input-prompt">
<!-- INSERT ANY VALID HTML HERE -->
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: Your answer may vary, but here is a sample.</p>
<p>-10, -100, -1000, -10,000. For example: The absolute values of the quotients are the same, but the sign is
opposite..</p>
</div>
</div>
<br>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<p id="fs-id1168345252532">3. In middle school, you used tape diagrams to represent division. This tape diagram shows that 6÷2=3<br></p>
<p id="fs-id1168345252532"><img alt="Tape diagram, 3 parts, each marked 2." class="img-fluid atto_image_button_text-bottom" height="20" src="https://k12.openstax.org/contents/raise/resources/2fa977cee72bd812dd16a0e13b6216fa87f3f171" width="100"><br>
</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="eventShow3" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<div class="os-raise-ib-cta-content">
<p id="fs-id1168345252532">Draw a tape diagram that shows why \(6\div\frac{1}{2}=12\). </p>
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
After you have drawn your answer, select the <strong>solution</strong> button to compare your work.<br>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="eventShow3">
<p id="fs-id1168345252532"><img alt="A tape diagram showing twelve groups of 1/2" class="img-fluid atto_image_button_text-bottom" height="51" src="https://k12.openstax.org/contents/raise/resources/b924c2718f08db2f7af4d4bbf553e11ecf06feee" width="252"></p>
</div>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="76b01fd0-ad76-4508-9959-8fb05b2c6aac" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<div class="os-raise-ib-input-content">
<p>4. Try to draw a tape diagram that represents 6÷0. Explain why this is so difficult.<br></p>
</div>
<div class="os-raise-ib-input-prompt">
<!-- INSERT ANY VALID HTML HERE -->
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: Your answer may vary, but here is a sample.</p>
<p>The zero takes up no space, so no matter how many copies we have it could never reach 6.</p>
</div>
</div>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="61798730-af10-4e2a-91da-c469c7e8fd41" data-schema-version="1.0" data-wait-for-event="eventShowMore1">
<div class="os-raise-ib-input-ack">
</div>
</div>