Lesson 7 More or Less Practice Understanding

Jump Start

Graph the solution set for each of the following single-variable inequalities by labeling the lines and shading the solutions:

1.

$2 x + 3 \leq 7$

2.

$2 x + 3 \geq 7$

3.

$2 x + 3 < 7$

4.

Since all of the coefficients are the same, what accounts for the difference in the solution sets?

Learning Focus

Solve systems of linear inequalities.

How do I interpret inequality signs, such as $>$ , $<$ , $\geq$ , and $\leq$ when determining what to shade as a solution set to an inequality?

Open Up the Math: Launch, Explore, Discuss

Solve the following systems of inequalities:

1.

${\begin{matrix} - 5 x + 3 y \leq 45 \\ 2 x + 3 y > 24 \end{matrix}$

2.

${\begin{matrix} - 10 x + 6 y \leq 90 \\ 6 x + 9 y > 36 \end{matrix}$

3.

Is the point $(- 3, 10)$ a solution to the system in problem 1? Why or why not?

4.

Write the system of inequalities whose solution set is shown:

5.

Amanda is examining Frank’s work on problem 4, when she exclaims, “You have written all of your inequalities backwards. The solution set to your system would look like this.”

What do you think about Amanda’s statement?

Pause and Reflect

Carlos and Clarita have found a cat food that seems to appeal to even the most finicky of cats, Figaro Flakes. They want to mix it with a less expensive cat food, Tabitha Tidbits, to make an affordable, but tasty cat food.

Tabitha Tidbits contains $4$ grams of protein and $6$ grams of fat per scoop. Figaro Flakes contains $12$ grams of protein and $4$ grams of fat per scoop. Carlos wants to make a meal for cats that contains at least $8$ grams of protein and no more than $6$ grams of fat per scoop.

6.

Write and solve a system of inequalities that Carlos can use to determine possible combinations of Tabitha Tidbits and Figaro Flakes that will satisfy both of these constraints.

7.

Based on your work, suggest at least 3 different “recipes” using each type of cat food that meets Carlos’s nutritional goals. For example, would $1$ scoop of Tabitha Tidbits and $\frac{1}{2}$ scoop of Figaro Flakes be an acceptable meal?

Ready for More?

Write a system of inequalities whose solution set would be a region of the plane that forms some geometric shape, such as a square, a rhombus, a parallelogram, or a trapezoid.

Takeaways

Describe a procedure for modeling a context using a system of inequalities as constraints:

Lesson Summary

In this lesson, we learned how to differentiate between linear inequalities in two variables written as strict inequalities, inequalities with $<$ or $>$ signs, and other inequalities written with $\leq$ or $\geq$ . We also practiced finding the solution set for systems of linear inequalities.

Retrieval

Substitute the given value into the equation to find the value of the variable that is not provided.

1.

$10 x - 9 y = 74; x = 2$

2.

$- 2 x + 9 y = 16; y = 4$

Graph each inequality.

3.

$y \leq 3 x - 4$

4.

$y \leq - 2 x + 3$