Representing Situations with Inequalities: IM Alg1.2.18
[url=https://im.kendallhunt.com/HS/teachers/1/2/18/preparation.html]“Representing Situations with Inequalities”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.2.18 Lesson
- IM Alg1.2.18 Lesson: Representing Situations with Inequalities
- Alg1.2.18 Practice
- IM Alg1.2.18 Practice: Representing Situations with Inequalities
IM Alg1.2.18 Lesson: Representing Situations with Inequalities
Match each inequality to the meaning of a symbol within it.
Is 25 a solution to any of the inequalities? Which one(s)?
Is 40 a solution to any of the inequalities? Which one(s)?
Is 30 a solution to any of the inequalities? Which one(s)?
Seniors in a student council of a high school are trying to come up with a budget for the Senior Ball.
Here is some information they have gathered:[list][*]Last year, 120 people attended. It was a success and is expected to be even bigger this year. Anywhere up to 200 people might attend.[/*][*]There needs to be at least 1 chaperone for every 20 students.[/*][*]The ticket price can not exceed $20 per person.[/*][*]The revenue from ticket sales needs to cover the cost of the meals and entertainment, and also make a profit of at least $200 to be contributed to the school.[/*][/list][size=150]Here are some inequalities the seniors wrote about the situation. Each letter stands for one quantity in the situation. Determine what is meant by each letter.[br][list][*][math]t\le20[/math][/*][/list][/size]
[list][*][math]120\le p\le200[/math][/*][/list]
[list][*][math]pt-m\ge200[/math][/*][/list]
[list][*][math]c\ge\frac{p}{20}[/math][/*][/list]
Kiran says we should add the constraint t ≥ 0.
What is the reasoning behind this constraint?[br]
What other "natural constraint" like this should be added?[br]
An elevator car in a skyscraper can hold at most 15 people. For safety reasons, each car can carry a maximum of 1,500 kg. On average, an adult weighs 70 kg and a child weighs 35 kg. Assume that each person carries 4 kg of gear with them.
Write as many equations and inequalities as you can think of to represent the constraints in this situation. Be sure to specify the meaning of any letters that you use. (Avoid using the letters [math]z[/math], [math]m[/math], or [math]g.[/math])[br]
[size=150]Trade your work with a partner and read each other's equations and inequalities.[/size][br][list][*]Explain to your partner what you think their statements mean, and listen to their explanation of yours.[/*][*]Make adjustments to your equations and inequalities so that they are communicated more clearly.[br][br][/*][/list]
[size=150]Rewrite your equations and inequalities so that they would work for a different building where:[/size][br][br][list][*]an elevator car can hold at most [math]z[/math] people[/*][/list]
[list][*]each car can carry a maximum of [math]m[/math] kilograms[br][br][/*][/list][br]
[list][*]each person carries [math]g[/math] kg of gear[/*][/list]
IM Alg1.2.18 Practice: Representing Situations with Inequalities
[size=150]Tyler goes to the store. His budget is $125. Which inequality represents [math]x[/math], the amount in dollars Tyler can spend at the store?[/size]
Jada is making lemonade for a get-together with her friends.
[size=150]She expects a total of 5 to 8 people to be there (including herself). She plans to prepare 2 cups of lemonade for each person.[br][br]The lemonade recipe calls for 4 scoops of lemonade powder for each quart of water. Each quart is equivalent to 4 cups.[/size][br][br]Let [math]n[/math] represent the number of people at the get-together, [math]c[/math] the number of cups of water, [math]\ell[/math] the number of scoops of lemonade powder.[br][br]Select [b]all[/b] the mathematical statements that represent the quantities and constraints in the situation.
A doctor sees between 7 and 12 patients each day.
[size=150]On Mondays and Tuesdays, the appointment times are 15 minutes. On Wednesdays and Thursdays, they are 30 minutes. On Fridays, they are one hour long. The doctor works for no more than 8 hours a day.[/size][br][br]Here are some inequalities that represent this situation.[br][br][table][tr][td][math]0.25\le y\le1[/math][/td][td][math]7\le x\le12[/math][/td][td][math]xy\le8[/math][/td][/tr][/table][br]What does each variable represent?[br]
What does the expression [math]xy[/math] in the last inequality mean in this situation?
Han wants to build a dog house. He makes a list of the materials needed:
[list][*]At least 60 square feet of plywood for the surfaces[/*][*]At least 36 feet of wood planks for the frame of the dog house [/*][*]Between 1 and 2 quarts of paint[/*][/list]Han's budget is $65. Plywood costs $0.70 per square foot, planks of wood cost $0.10 per foot, and paint costs $8 per quart.[br][br]Write inequalities to represent the material constraints and cost contraints in this situation. Be sure to specify what your variables represent.
[size=150]The equation [math]V=\frac{1}{3}\pi r^2h[/math] represents the volume of a cone, where [math]r[/math] is the radius of the cone and [math]h[/math] is the height of the cone.[br][br][/size]Which equation is solved for the height of the cone?
Solve each system of equations without graphing.
[math]\displaystyle \begin{cases} 2x+3y=5 \\ 2x+4y=9 \\ \end{cases}[/math]
[math]\begin{cases} \frac23x+y=\frac73\\ \frac23x-y=1 \\ \end{cases}[/math]
There is a pair of x and y values that make each equation true in this system of equations:
[size=150]There is a pair of [math]x[/math] and [math]y[/math] values that make each equation true in this system of equations:[/size][br][math]\begin{cases} 5x + 3y = 8 \\ \text 4x + 7y = 34 \\ \end{cases}[/math][br][br]Explain why the same pair of values also make [math]9x+10y=42[/math] true.
Which ordered pair is a solution to this system of equations?
[math]\begin{cases} 7x+5y=59 \\ 3x-9y=159 \\ \end{cases}[/math]
[size=150]Which equation has exactly one solution in common with the equation [math]y=6x-2[/math]?[/size]
How many solutions does this system of equations have? Explain how you know.
[math]\displaystyle \begin{cases} y=\text-4x+3\\ 2x+8y=10\\ \end{cases}[/math]