Lesson 26

Testing Points to Solve Inequalities

These materials, when encountered before Algebra 1, Unit 2, Lesson 26 support success in that lesson.

26.1: Math Talk: Solving Equations (10 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for reasoning about equations that have expressions in parentheses. These understandings help students develop fluency and will be helpful in the associated Algebra 1 lesson when they solve equations as part of solving inequalities.

In this activity, students have an opportunity to notice and make use of structure (MP7), because they first solve equations in the form \(a = mx + b\), and then in the form \(a = m(x – h) + b\), and they have an opportunity to apply their strategies for solving the first equation to the second.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Solve each equation mentally.

\(3x+5=14\)

\(3(x-1)+5=14\)

\(3x-3+5=14\)

\(3(1-x)+5=14\)

Student Response

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Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?

If no student mentions it, ask, “Did the thinking you did for any of the previous problems help you with this problem?”

26.2: Character Chat (20 minutes)

Activity

The mathematical purpose of this lesson is to help students reason about why testing points is an important step in solving inequalities, and how to do so purposefully and efficiently. This helps make explicit the reasoning behind some of the work students are doing in the associated Algebra 1 lesson.

Launch

Display Andre’s work for all to see. Ask students what they notice, and what they wonder about Andre’s work. Give students an opportunity to work on the questions about Andre’s work.

If students are struggling to make sense of Mai’s work, encourage students to notice and wonder about it in small groups, or pause the class when most students are ready to move on to Mai’s work. Display Mai’s work for all to see. Ask students what they notice, and what they wonder about Mai’s work. Explain that Mai has made an error, and ask them to notice and wonder again. Give students an opportunity to work on the questions about Mai’s work.

Student Facing

Andre is working on \(\frac{5x}{3} - 1 < \frac23\). He figured out that when \(x = 1, \frac{5(1)}{3} - 1 = \frac23\). He tested all these points:

  • When \(x = \text-1, \frac{5(\text-1)}{3} - 1 = \frac{\text-8}{3}, \frac{\text-8}{3} < \frac23\)
  • When \(x = 0, \frac{5(0)}{3} - 1 = \text-1, \text-1 < \frac23\)
  • When \(x = 2, \frac{5(2)}{3} - 1 = \frac73, \frac73 > \frac23\)
  • When \(x = 3, \frac{5(3)}{3} - 1 = 4, 4 > \frac23\)

Based on these results, Andre determines that solutions for \(x\) should include -1 and 0, but not 2 or 3.

  1. Andre is frustrated with how much computation he had to do. What advice would you give him about how many numbers to test and which ones to test?
  2. Mai was trying to solve \(10 - 3x > 7\). She saw that when \(10 - 3(1) = 7\). She reasoned, “Because the problem has a greater than sign, I wrote \(x > 1\).” Mai skipped the step of testing points, and that led to an error.
    1. Help Mai test points to determine the correct solution to the inequality.
    2. Explain to Mai what went wrong with her reasoning.

Student Response

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Activity Synthesis

The purpose of this discussion is to help students continue to practice reasoning about what is going on mathematically when they test points to help them solve inequalities. Students should be beginning to recognize the value of solving the related equation and testing a point to the left or right, and picking those points strategically.

Here are some questions for discussion:

Blank number line.

Display a number line for all to see. Ask students:

  • "What would Andre see if he color-coded the points he tested on his number line?" (He should see that all of the values to the left of 1 on the number line are all the same color and all the values to the right of 1 on the number line are the other color.)
  • "Once Andre plotted and color-coded 1, and one other point, would he be able to predict what color the other points would be?" (Yes. All of the points on the same side of 1 on the number line as the point checked should be the same color and the other side of 1 on the number line would be another color.)
  • "If Andre had made a mistake and \(x = 1\) was not the correct solution to \(\frac{5x}{3} - 1 = \frac23\), how could testing more than 1 point help Andre spot his mistake?" (If he notices that two points on the same side of 1 on the number line has different colors, that would be a clue that he made a mistake.)
  • "What was Mai’s error in her thinking?" (The direction of the inequality from the original question is the same as the direction of the inequality in the solution.)
  • "What might have helped Mai fix her mistake?" (Testing points on one side or the other of the solution to the equation would be helpful to determine the direction of the inequality.)

Instruct students to explain to a partner how Andre could have figured out the solutions to the inequality without having to test all the points.

Instruct students to tell their partner how they could use Mai’s experience to convince someone that it is a good idea to test more than one point.

26.3: Error! (15 minutes)

Activity

In this activity, students get a chance to practice spotting and correcting common errors when solving equations.

They can use the fact that they are told each equation has an error in it to practice checking their thinking about solving equations.

The practice will pay off when they solve equations and inequalities in the associated Algebra 1 lesson.

Launch

Explain to students that each example has an error in it, and that they should spot the error and explain what each character should have done differently.

Student Facing

Each of these solutions has something wrong. Circle the place that is wrong and write a correction.

  1. \(\begin{align} 2x + 3 &= 5x – 4 \\ 5x &= 5x – 4 \\ 0 &= \text-4 \\ \end{align}\)
  2. \(\begin{align} 5x + 4 &= 10 - 5x \\ 4 &= 10 \\ \end{align}\)
  3. \(\begin{align} 2x + 8 &= 2x + 100 \\ 4x + 8 &= 100 \\ x + 2 &= 50 \\ x &= 48 \\ \end{align}\)
  4. \(\begin{align} 5x + 50 &= 20x \\ 50 &= 25x \\ 2 &= x \\ \end{align}\)
  5. \(\begin{align} 2(x + 8) &= 16 \\ 2x + 16 &= 16 \\ 2x &= 0 \\ \end{align}\)
    No solution
  6. \(\begin{align} (x + 3) + 5 &= 5 \\ x + 3 &= 0 \\ x = 3 \\ \end{align}\)

Student Response

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Activity Synthesis

The main goal of this lesson is for students to continue thinking about how solving inequalities is a process of reasoning, and the role of testing points in that reasoning.

An additional goal is to practice error analysis and spotting mistakes when solving equations and inequalities.

Possible questions for reflection:

  • "How does testing points in an inequality help you tell if you made a mistake when solving the inequality?" (If a value makes the solution true, but not the original question, then there is likely a mistake in solving the inequality.)
  • "If you solved the related equation correctly, why would you only need to test one point to find out which values are solutions to the inequality?" (The solution to an equation related to an inequality divides the number line into two sections. All points on one side of the equation's solution make the inequality more extreme in one direction or the other. Finding a point on one side of the number line that makes the inequality true should indicate that all points on that side of the number line are solutions to the inequality.)
  • "If you made a mistake solving the equation, how could testing more than one point help you find your mistake?" (If you find two points on the same side of the number line such that one makes the inequality true and the other does not, then you likely made a mistake solving the related equation.)
  • "What other strategies could you use to help you spot mistakes in your work?" (Examine the algebra for common mistakes or check with a partner.)
  • "Are there any errors you find harder to spot? What could help you look for those errors in your own work or when helping a friend?"