Lesson 8 Quadratic Quandaries Practice Understanding

Learning Focus

Solve quadratic inequalities both graphically and algebraically.

Interpret solutions to quadratic inequalities that arise from context.

Solve a system of equations that contains both a quadratic and linear equation.

How can we use our understanding of solving quadratic equations and graphing quadratic functions to solve quadratic inequalities?

What do our solutions mean in the context of the problem?

How do we identify and write all the solutions?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita have a brilliant idea for how they will earn money this summer. Since the community in which they live includes many high schools, a couple of universities, and even some professional sports teams, it seems that everyone has a favorite team they like to root for. In Carlos’s and Clarita’s neighborhood, these rivalries take on special meaning, since many of the neighbors support different teams. They have observed that their neighbors often display handmade posters and other items to make their support of their favorite team known. The twins believe they can get people in the neighborhood to buy into their new project: painting team logos on curbs or driveways.

For a small fee, Carlos and Clarita will paint the logo of a team on a neighbor’s curb, next to their house number. For a larger fee, the twins will paint a mascot on the driveway. Carlos and Clarita have designed stencils to make the painting easier and they have priced the cost of supplies. They have also surveyed neighbors to get a sense of how many people in the community might be interested in purchasing their service.

Surveys show the twins can sell $100$ driveway mascots at a cost of $$ 20$ , and they will sell $10$ fewer mascots for each additional $$ 5$ they charge.
The twins estimate that the cost of supplies will be $$ 250$ and they would like to make $$ 2, 000$ in profit from selling driveway mascots. Therefore, they will need to collect $$ 2, 250$ in revenue.

This information led Carlos and Clarita to write and solve the quadratic equation:

$(100 - 10 x) (20 + 5 x) = 2, 250$

1.

Solve this quadratic equation for $x$ .

2.

What do your solutions for $x$ mean in terms of the story context?

3.

How would your solution change if this had been the question Carlos and Clarita had asked: “How much should we charge if we want to collect at least $$ 2, 250$ in revenue?”

4.

What about this question: “How much should we charge if we want to maximize our revenue?”

As you probably observed, the situation represented in problem 3 didn’t have just one solution, since there are many different prices the twins can charge to collect more than $$ 2, 250$ in revenue. Sometimes our questions lead to quadratic inequalities rather than quadratic equations.

Here is another quadratic inequality generated by Carlos and Clarita’s business plans:

5.

Carlos and Clarita want to design a logo that requires less than $48 {in}^{2}$ of paint and that fits inside a rectangle that is $8$ inches longer than it is wide. What are the possible dimensions of the rectangular logo?

Again, problem 5 has multiple answers, and those answers are restricted by the context. Let’s examine the inequality you wrote for problem 5, but not restricted by the context.

6.

What are the solutions to the inequality $x (x + 8) < 48$ ?

7.

How might you support your answer to problem 6 with a graph or a table?

Here are some more quadratic inequalities without contexts. Show how you might use a graph, along with algebra, to solve each of them.

8.

$x^{2} + 3 x - 10 \geq 0$

9.

$2 x^{2} - 5 x < 12$

10.

$x^{2} - 4 \leq 4 x + 1$

Carlos and Clarita both used algebra and a graph to solve problem 10, but they both did so in different ways. Illustrate each of their methods with a graph and with algebra.

11.

Carlos: “I rewrote the inequality to get $0$ on one side and a factored form on the other. I found the zeroes for each of my factors. To decide what values of $x$ made sense in the inequality, I also sketched a graph of the quadratic function that was related to the quadratic expression in my inequality. I shaded solutions for $x$ based on the information from my graph.”

12.

Clarita: “I graphed a linear function and a quadratic function related to the linear and quadratic expressions in the inequality. From the graph, I could estimate the points of intersection, but to be more exact, I solved the quadratic equation $x^{2} - 4 = 4 x + 1$ by writing an equivalent equation that had $0$ on one side. Once I knew the $x$ -values for the points of intersection in the graph, I could shade solutions for $x$ that made the inequality true.”

When Carlos examined Clarita’s method he observed, “What you’re really doing is writing two equations, $y_{1} = x^{2} - x$ and $y_{2} = 4 x + 1$ . When you find the intersection of the graphs, you’re also finding the solution to the system of equations.”

Clarita is impressed with the connection that Carlos made to solving a system of equations. You might be impressed if you try it.

13.

Use Clarita’s method to solve the system of equations graphically.

${\begin{cases} y = x^{2} - 6 x + 11 \\ y = x + 1 \end{cases}$

14.

Verify your solutions using an algebraic method.

15.

Solve this system of equation using the method of your choice. Show and explain your work below. A grid has been provided to use if you choose a graphical method.

${\begin{cases} y = x^{2} + 6 x + 6 \\ y = - 2 x - 6 \end{cases}$

Algebraically:

Graphically:

Ready for More?

Devise a strategy based on your work with quadratic inequalities that could be used to solve this cubic inequality with three factors:

$(x - 4) (x + 4) (x + 8) \leq 0$

Takeaways

Solving quadratic inequalities:

Vocabulary

quadratic inequality
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we developed a strategy for solving quadratic inequalities. The procedure involves solving the related quadratic equation and then using the graph or testing values to find the intervals that are solutions to the inequality. If the inequality represents a real context, the solutions must be interpreted so that they fit the situation.

Retrieval

1.

Factor completely. $- 6 x^{2} - 15 x + 9$

2.

Write in vertex form. $f (x) = (x + 3) (x - 7)$