4.2 Graphing Linear Inequalities

Posljednje ažuriranje over 1 year ago

Napomena autora:

Instrukcije

Intro - Warm-Up

Explore It

Learn It

4.2 Graphing Linear Inequalities

Posljednje ažuriranje over 1 year ago

Napomena autora:

OBJECTIVES & STANDARDS

Math Objectives

Graph linear inequalities and identify solutions.
Write linear inequalities from word problems.
Interpret solutions to inequalities in terms of a real-world context.

Common Core Math Standards

Personal Finance Objectives

Model budgeting constraints based on two sources of income.

National Standards for Personal Financial Education

Earning Income

11a: Evaluate the benefits and costs of gig employment, such as driving for a cab or delivery service.

DISTRIBUTION & PLANNING

Distribute to students

Instrukcije

OBJECTIVES & STANDARDS

Math Objectives

Graph linear inequalities and identify solutions.
Write linear inequalities from word problems.
Interpret solutions to inequalities in terms of a real-world context.

Common Core Math Standards

Personal Finance Objectives

Model budgeting constraints based on two sources of income.

National Standards for Personal Financial Education

Earning Income

11a: Evaluate the benefits and costs of gig employment, such as driving for a cab or delivery service.

DISTRIBUTION & PLANNING

Distribute to students

Intro - Warm-Up

INTERACTIVE: The Uber Game

We all encounter different limitations as we go through the day - what we can and can’t do.

We have to make decisions based on those limitations. How do we spend our limited time, money, and energy? Play the game and answer the reflection questions.

Which choices were the hardest to make in the game? Why?

While playing the game, what limitations or constraints did you face?

At the end of the game, how much did you earn per hour on average?

Write an equation representing how many hours you have to work (x) in order to earn $1,000 after paying expenses (y).

Adjust your equation to reflect that you want to earn MORE than $1000, not exactly $1000.

Explore It

Linear Inequalities

In the previous activity, you made trade-offs based on real-world constraints. We can represent these types of constraints using linear inequalities. Explore the example inequality and graph below, before learning how to graph real-world linear inequalities in the next activity.

You’re looking for two numbers that add up to less than 35. Identify three possible sets of numbers that add up to less than 35.

The graph below represents that inequality.
What do you notice about this graph? How is it similar or different from graphs you’ve seen of linear equations?

Learn It

Graph the linear inequality y ≥ 15x in Desmos.

Note: This represents Cole’s earnings working as a lead barista, where he earns at least $15 per hour, depending on tips.

Desmos #1:
ACTIVITY: Point Collector- Lines
Follow your teacher’s instructions to complete this Desmos activity. Then, answer the question.
When graphing a linear inequality, how do you know which direction to shade?

Contextualizing Linear Inequalities
You know how to write linear equations from word problems. Now, you’ll adapt that skill to write linear inequalities. Look out for key phrases that will tell you which sign to use, like “at least”, “a minimum” and “no more than”. Notes in Book

Match each inequality with the situation it represents.

Stavka koja se može prevući		Odgovarajuća stavka
A $50 one-time expense to purchase necessary supplies and earnings of more than $15 per hour		y ≥ 15x + 50
A $50 one-time expense to purchase necessary supplies and maximum earnings of $15 per hour		y ≤ 15x + 50
A transportation budget of at most $50 to spend on $1 bus rides and $15 cab rides		y > 15x - 50
A yard sale earns more than $50 selling $15 clothing items and $1 books		y ≤ 15x - 50
A $50 one-time signing bonus and earnings up to $15 per hour		x + 15y ≤ 50
A $50 one-time signing bonus and earnings of at least $15 per hour		x + 15y > 50

Kevin works two part-time jobs and wants to earn a total of at least $504 per week. He earns an average of $12 per hour delivering food and $14 per hour working retail.

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c. The point (42, 0) is a solution to the inequality. What does that point represent in this context?

d. Is that point an optimal real-world solution? Why or why not?

e. Identify another possible combination of hours that Kevin could work.

Sana works part-time in event management earning $20 per hour. She also does freelance carpentry projects for $25 per hour. She wants to earn more than $750 per week.

Write an inequality to represent the scenario.
Graph the inequality.
Identify one possible solution to the inequality. Is it reasonable in this context? Why or why not?
Sana covers her basic expenses with $750, but wants to save up for a vacation. How much extra would she earn if she worked 30 hours in event management and 15 hours in carpentry?
If Sana can’t find any hours of carpentry work, at least how many hours would she need to work in event management to meet her basic needs?
What is the fewest number of hours that Sana could work and still earn more than $750? Justify your response.

Identify one possible solution to the inequality. Is it reasonable in this context? Why or why not?

Sana covers her basic expenses with $750, but wants to save up for a vacation. How much extra would she earn if she worked 30 hours in event management and 15 hours in carpentry?

If Sana can’t find any hours of carpentry work, at least how many hours would she need to work in event management to meet her basic needs?

Quentin works as a used car salesman, where he earns $8 per hour and an average commission of $260 per car sold. He wants to earn at least $1040 per week.

Write an inequality to represent this scenario, where x represents hours worked and y represents cars sold.
Graph the Inequality
The point (35, 3) is a solution. What does it represent in this context?
If Quentin only sells two cars that week, at least how many hours would he need to work?
If Quentin wants to work a standard 40-hour workweek, at least how many cars would he need to sell?
Identify one point that IS a solution but is NOT reasonable in the real-world context. Justify your response.

4.2 Graphing Linear Inequalities

4.2 Graphing Linear Inequalities

Which choices were the hardest to make in the game? Why?

While playing the game, what limitations or constraints did you face?

At the end of the game, how much did you earn per hour on average?

Write an equation representing how many hours you have to work (x) in order to earn $1,000 after paying expenses (y).

Adjust your equation to reflect that you want to earn MORE than $1000, not exactly $1000.

You’re looking for two numbers that add up to less than 35. Identify three possible sets of numbers that add up to less than 35.

The graph below represents that inequality.What do you notice about this graph? How is it similar or different from graphs you’ve seen of linear equations?

Desmos #1: ACTIVITY: Point Collector- Lines Follow your teacher’s instructions to complete this Desmos activity. Then, answer the question.When graphing a linear inequality, how do you know which direction to shade?

Contextualizing Linear InequalitiesYou know how to write linear equations from word problems. Now, you’ll adapt that skill to write linear inequalities. Look out for key phrases that will tell you which sign to use, like “at least”, “a minimum” and “no more than”. *Notes in Book*

Match each inequality with the situation it represents.

c. The point (42, 0) is a solution to the inequality. What does that point represent in this context?

d. Is that point an optimal real-world solution? Why or why not?

e. Identify another possible combination of hours that Kevin could work.

Identify one possible solution to the inequality. Is it reasonable in this context? Why or why not?

Sana covers her basic expenses with $750, but wants to save up for a vacation. How much extra would she earn if she worked 30 hours in event management and 15 hours in carpentry?

If Sana can’t find any hours of carpentry work, at least how many hours would she need to work in event management to meet her basic needs?

The point (35, 3) is a solution. What does it represent in this context?

If Quentin only sells two cars that week, at least how many hours would he need to work?

If Quentin wants to work a standard 40-hour workweek, at least how many cars would he need to sell?

Which choices were the hardest to make in the game? Why?

While playing the game, what limitations or constraints did you face?

At the end of the game, how much did you earn per hour on average?

Write an equation representing how many hours you have to work (x) in order to earn $1,000 after paying expenses (y).

Adjust your equation to reflect that you want to earn MORE than $1000, not exactly $1000.

You’re looking for two numbers that add up to less than 35. Identify three possible sets of numbers that add up to less than 35.

The graph below represents that inequality.What do you notice about this graph? How is it similar or different from graphs you’ve seen of linear equations?

Mark the three solutions from Question 1 on the graph of the inequality in the Show Your Work below. What do you notice?

What do you think the shaded region represents on the graph?

Mark the point (19, 16) on the graph.Is (19, 16) a solution to the inequality x + y < 35? Why or why not?Why do you think the line in the graph is dashed, not solid?

Summarize: What is the difference between a linear inequality and a linear equation?

1. Identify key features In slope-intercept form, find the y-intercept and slope. In standard form, find the x-intercept and y-intercept.

2. Graph the lineIf the inequality uses ≤ or ≥, the line is solid. If it uses < or >, the line is dashed.

3. Test point and shade Choose a point and plug it into the inequality. If the point makes the statement true, shade that side of the line. If not, shade the other side.

Identify solution(s)

Jordan wants to earn more than $120 each week. They earn $12 per hour as a lifeguard and $8 per hour babysitting. Graph the linear inequality representing this scenario: 12x + 8y > 120

Identify key featuresIntercepts and/or slope

2. Graph the lineIs the line Dashed or solid?

3, Test point and shade Choose a point and plug it into the inequality. If the point makes the statement true, shade that side of the line. If not, shade the other side.

4. Identify solution(s)

Mark the point (7, 7) on the graph from question 17. Would Jordan earn enough money if they worked 7 hours lifeguarding and 7 hours babysitting? Justify your response.

Is (8, 3) a solution to this inequality? How do you know?

Desmos #1: ACTIVITY: Point Collector- Lines Follow your teacher’s instructions to complete this Desmos activity. Then, answer the question.When graphing a linear inequality, how do you know which direction to shade?

Contextualizing Linear InequalitiesYou know how to write linear equations from word problems. Now, you’ll adapt that skill to write linear inequalities. Look out for key phrases that will tell you which sign to use, like “at least”, “a minimum” and “no more than”. *Notes in Book*

Match each inequality with the situation it represents.

c. The point (42, 0) is a solution to the inequality. What does that point represent in this context?

d. Is that point an optimal real-world solution? Why or why not?

e. Identify another possible combination of hours that Kevin could work.

Identify one possible solution to the inequality. Is it reasonable in this context? Why or why not?

Sana covers her basic expenses with $750, but wants to save up for a vacation. How much extra would she earn if she worked 30 hours in event management and 15 hours in carpentry?

If Sana can’t find any hours of carpentry work, at least how many hours would she need to work in event management to meet her basic needs?

What is the fewest number of hours that Sana could work and still earn more than $750? Justify your response.

The point (35, 3) is a solution. What does it represent in this context?

If Quentin only sells two cars that week, at least how many hours would he need to work?

If Quentin wants to work a standard 40-hour workweek, at least how many cars would he need to sell?

Identify one point that IS a solution but is NOT reasonable in the real-world context. Justify your response.

Mark the three solutions from Question 1 on the graph of the inequality in the Show Your Work below. What do you notice?

What do you think the shaded region represents on the graph?

Mark the point (19, 16) on the graph.Is (19, 16) a solution to the inequality x + y < 35? Why or why not?Why do you think the line in the graph is dashed, not solid?

Summarize: What is the difference between a linear inequality and a linear equation?

1. Identify key features In slope-intercept form, find the y-intercept and slope. In standard form, find the x-intercept and y-intercept.

2. Graph the lineIf the inequality uses ≤ or ≥, the line is solid. If it uses < or >, the line is dashed.

3. Test point and shade Choose a point and plug it into the inequality. If the point makes the statement true, shade that side of the line. If not, shade the other side.

Identify solution(s)

Jordan wants to earn more than $120 each week. They earn $12 per hour as a lifeguard and $8 per hour babysitting. Graph the linear inequality representing this scenario: 12x + 8y > 120

Identify key featuresIntercepts and/or slope

2. Graph the lineIs the line Dashed or solid?

3, Test point and shade Choose a point and plug it into the inequality. If the point makes the statement true, shade that side of the line. If not, shade the other side.

4. Identify solution(s)

Mark the point (7, 7) on the graph from question 17. Would Jordan earn enough money if they worked 7 hours lifeguarding and 7 hours babysitting? Justify your response.

Is (8, 3) a solution to this inequality? How do you know?

What is the fewest number of hours that Sana could work and still earn more than $750? Justify your response.

Identify one point that IS a solution but is NOT reasonable in the real-world context. Justify your response.

The graph below represents that inequality.
What do you notice about this graph? How is it similar or different from graphs you’ve seen of linear equations?

Desmos #1:
ACTIVITY: Point Collector- Lines
Follow your teacher’s instructions to complete this Desmos activity. Then, answer the question.
When graphing a linear inequality, how do you know which direction to shade?

Contextualizing Linear Inequalities
You know how to write linear equations from word problems. Now, you’ll adapt that skill to write linear inequalities. Look out for key phrases that will tell you which sign to use, like “at least”, “a minimum” and “no more than”. Notes in Book

The graph below represents that inequality.
What do you notice about this graph? How is it similar or different from graphs you’ve seen of linear equations?

Mark the point (19, 16) on the graph.
Is (19, 16) a solution to the inequality x + y < 35? Why or why not?
Why do you think the line in the graph is dashed, not solid?

2. Graph the line
If the inequality uses ≤ or ≥, the line is solid. If it uses < or >, the line is dashed.

Identify key features
Intercepts and/or slope

2. Graph the line
Is the line Dashed or solid?

Desmos #1:
ACTIVITY: Point Collector- Lines
Follow your teacher’s instructions to complete this Desmos activity. Then, answer the question.
When graphing a linear inequality, how do you know which direction to shade?

Contextualizing Linear Inequalities
You know how to write linear equations from word problems. Now, you’ll adapt that skill to write linear inequalities. Look out for key phrases that will tell you which sign to use, like “at least”, “a minimum” and “no more than”. Notes in Book

Mark the point (19, 16) on the graph.
Is (19, 16) a solution to the inequality x + y < 35? Why or why not?
Why do you think the line in the graph is dashed, not solid?

2. Graph the line
If the inequality uses ≤ or ≥, the line is solid. If it uses < or >, the line is dashed.

Identify key features
Intercepts and/or slope

2. Graph the line
Is the line Dashed or solid?