Lesson 6 Taking Sides Practice Understanding

Learning Focus

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 “ $6$ less than a number is greater than $4$ .” I think that we should just follow the words and write: $6 - n > 4$ .

Serena: I don’t think that works because if $n$ is $20$ and you do $6$ less than that you get $20 - 6 = 14$ . I think we should write $n - 6 > 4$ .

Joaquin: Oh, you’re right. Then it makes sense that the solution will be $n > 10$ , which means we can choose any number greater than $10$ .

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.

1.

Joaquin and Serena are assigned to graph the inequality $x \geq - 7$ .

Joaquin thinks the graph should have an open dot at $- 7$ .

Serena thinks the graph should have a closed dot at $- 7$ .

Explain who is correct and why.

2.

Joaquin and Serena are looking at the problem $3 x + 1 > 0$ .

Serena says that the inequality is always true because multiplying a number by $3$ and then adding $1$ to it makes the number greater than $0$ .

Is she right? Explain why or why not.

3.

The word problem that Joaquin and Serena are working on says, “ $4$ greater than $x$ .”

Joaquin says that they should write: $4 > x$

Serena says they should write: $4 + x$

Explain who is correct and why.

4.

Joaquin is thinking hard about equations and inequalities and comes up with this idea:

If $45 + 47 = t$ , then $t = 45 + 47$ .

So, if $45 + 47 < t$ , then $t < 45 + 47$ .

Is he right? Explain why or why not.

5.

Joaquin’s question in problem 4 made Serena think about other similarities and differences in equations and inequalities. Serena wonders about the equation $- \frac{x}{3} = 4$ and the inequality $- \frac{x}{3} > 4$ . Explain to Serena ways that solving these two problems are alike and ways that they are different. How are the solutions to the problems alike and different?

6.

a.

Joaquin solved $- 15 q \leq 135$ by adding $15$ to each side of the inequality. Serena said that he was wrong. Who do you think is right and why?

b.

Joaquin’s solution was $q \leq 150$ . He checked his work by substituting $150$ for $q$ in the original inequality. Does this prove that Joaquin is right? Explain why or why not.

c.

Joaquin is still skeptical and believes he is right. Find a number that satisfies his solution but does not satisfy the original inequality.

7.

Serena is checking her work with Joaquin and finds that they disagree on a problem. Here’s what Serena wrote:

$3 x + 3 \leq - 2 x + 5$ $3 x \leq - 2 x + 2$ $x \leq 2$ Is she right? Explain why or why not.

8.

Joaquin and Serena are having trouble solving $- 4 (3 m - 1) \geq 2 (m + 3)$ .

Explain how they should solve the inequality, showing all the necessary steps and identifying the properties you would use.

9.

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?

a.

$4 s + 6 \geq 6 + 4 s$

b.

$3 r + 5 > 3 r - 2$

c.

$4 (n + 1) < 4 n - 3$

10.

The partners are given the literal inequality $a x + b > c$ to solve for $x$ . Joaquin says that he will solve it just like an equation. Serena says that he needs to be careful because if $a$ is a negative number, the solution will be different. What do you say? What are the solutions for the 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.

1.

$| x | < 5$

2.

$| x | \geq 5$

Takeaways

Similarities in Solving Inequalities and Equations:

Differences in Solving Inequalities and Equations:

Lesson Summary

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.

Retrieval

1.

Graph each set of equations and find the point where they intersect.

$y = x + 3$

$y = - 2 x - 6$

Point:

Solve each literal equation for the indicated variable.

2.

$A = π r^{2}$ Solve for $r$ .

3.

$y = m x + b$ Solve for $m$ .