Lesson 6 Taking Sides Practice Understanding
Understand similarities and differences in solving equations and inequalities.
Learn to avoid common errors and misunderstandings about inequalities.
What are some of the common misconceptions of inequalities?
How does having a deep understanding of what inequalities mean help to avoid errors?
Can an inequality have no solutions?
Open Up the Math: Launch, Explore, Discuss
Joaquin and Serena work together productively in their math class. They both contribute their thinking and when they disagree, they both give their reasons and decide together who is right. In their math class right now, they are working on inequalities. Recently, they had a discussion that went something like this:
Joaquin: The problem says that “
Serena: I don’t think that works because if
Joaquin: Oh, you’re right. Then it makes sense that the solution will be
The situations below are a few more of the disagreements and questions that Joaquin and Serena have. Your job is to decide how to answer their questions, decide who is right, and give a mathematical explanation of your reasoning.
Joaquin and Serena are assigned to graph the inequality
Joaquin thinks the graph should have an open dot at
Serena thinks the graph should have a closed dot at
Explain who is correct and why.
Joaquin and Serena are looking at the problem
Serena says that the inequality is always true because multiplying a number by
Is she right? Explain why or why not.
The word problem that Joaquin and Serena are working on says, “
Joaquin says that they should write:
Serena says they should write:
Explain who is correct and why.
Joaquin is thinking hard about equations and inequalities and comes up with this idea:
Is he right? Explain why or why not.
Joaquin’s question in problem 4 made Serena think about other similarities and differences in equations and inequalities. Serena wonders about the equation
Joaquin’s solution was
Joaquin is still skeptical and believes he is right. Find a number that satisfies his solution but does not satisfy the original inequality.
Serena is checking her work with Joaquin and finds that they disagree on a problem. Here’s what Serena wrote:
Joaquin and Serena are having trouble solving
Explain how they should solve the inequality, showing all the necessary steps and identifying the properties you would use.
Joaquin and Serena know that some equations are true for any value of the variable and some equations are never true, no matter what value is chosen for the variable. They are wondering about inequalities. What could you tell them about the following inequalities? Do they have solutions? What are they? How would you graph their solutions on a number line?
The partners are given the literal inequality
Ready for More?
Use your reasoning skills to solve the following inequalities. Write your solutions in both interval and set notation and graph on the number line.
Similarities in Solving Inequalities and Equations:
Differences in Solving Inequalities and Equations:
In this lesson, we examined common mistakes and misconceptions about inequalities. We analyzed the similarities and differences in solving equations and inequalities, determining which properties applied to both and which properties were different for inequalities.
Graph each set of equations and find the point where they intersect.
Solve each literal equation for the indicated variable.