IM Alg I Unit 2 Lesson 19: Solutions to Inequalities in One Variable

Last updated about 3 years ago

26 questions

IM Alg I Unit 2 Lesson 19: Solutions to Inequalities in One Variable

Last updated about 3 years ago

26 questions

Lesson 19: Solutions to Inequalities in One Variable

Learning Targets 
I can graph the solution to an inequality in one variable. 
I can solve one-variable inequalities and interpret the solutions in terms of the situation. 
I understand that the solution to an inequality is a range of values (such as x>7) that make the inequality true.

Student Learning Goals: Let’s find and interpret solutions to inequalities in one variable.

Rewrite the student learning goals in your own words.

19.1 Find a Value, Any Value

If needed, you can use the number line to help them in reasoning about the inequalities.

Write some solutions to the inequality y\leq9.2. Be prepared to explain what makes a value a solution to this inequality.

Write one solution to the inequality 7(3-x)>14. Explain your reasoning.

How did you know that the value you chose is a solution?

What do you notice about all the points that are on the line?

On the number line, we can see that the solutions are values that are less than 1. All these values form the solution set to the inequality. Is there a way to write the solution set concisely, without using the number line and without writing out all the numbers less than 1?

Does the solution set have anything to do with the solution to the equation 7(3-x)=14?

Why does the solution set to the inequality 7(3-x)>14 involve numbers less than 1?

19.2 Off to an Orchard

A teacher is choosing between two options for a class field trip to an orchard.
At Orchard A, admission costs $9 per person and 3 chaperones are required. 
At Orchard B, the cost is $10 per person, but only 1 chaperone is required. 
At each orchard, the same price applies to both chaperones and students.

Which orchard would be cheaper to visit if the class has 8 students?

Which orchard would be cheaper to visit if the class has 12 students?

Which orchard would be cheaper to visit if the class has 30 students?

To help her compare the cost of her two options, the teacher first writes the equation 9(n+3)=10(n+1), and then she writes the inequality 9(n+3)<10(n+1).

What does n represent in each statement?

In this situation, what does the equation 9(n+3)=10(n+1) mean?

What does the solution to the inequality 9(n+3)<10(n+1) tell us?

19.3 Part-Time Work

To help pay for his tuition, a college student plans to work in the evenings and on weekends. He has been offered two part-time jobs: working in the guest-services department at a hotel and waiting tables at a popular restaurant.

The job at the hotel pays $18 an hour and offers $33 in transportation allowance per month. 
The job at the restaurant pays $7.50 an hour plus tips. The entire waitstaff typically collects about $50 in tips each hour. Tips are divided equally among the 4 waitstaff members who share a shift.

The equation 7.50h+\frac{50}{4}h=18h+33 represents a possible constraint about a situation.

Solve the equation and check your solution.

Here is a graph on a number line. Put a scale on the number line so that the point marked with a circle represents the solution to the equation.

Does one job pay better if the student works fewer hours than the solution you found earlier? If so, which job?

Does one job pay better if the student works more hours than the solution you found earlier? If so, which job?

Here are two inequalities and two graphs that represent the solutions to the inequalities. Match each inequality with a graph that shows its solution.

Draggable item		Corresponding Item
7.50h+\frac{50}{4}h<18h+33
7.50h+\frac{50}{4}h>18h+33

19.4 Equality and Inequality

Solve this equation and check your solution: -\frac{4(x+3)}{5}=4x-12

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values less than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values greater than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose 2 for x. Is it a solution?

Graph the solution to the inequality.

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

Lesson Synthesis

How does solving the equation 4x-3=12(x+3) help with solving the inequality 4x-3\geq12(x+3)

19.6 Seeking Solutions

IM Alg I Unit 2 Lesson 19: Solutions to Inequalities in One Variable

IM Alg I Unit 2 Lesson 19: Solutions to Inequalities in One Variable

Lesson 19: Solutions to Inequalities in One Variable

Rewrite the student learning goals in your own words.

Write some solutions to the inequality y\leq9.2. Be prepared to explain what makes a value a solution to this inequality.

Write one solution to the inequality 7(3-x)>14. Explain your reasoning.

How did you know that the value you chose is a solution?

What do you notice about all the points that are on the line?

On the number line, we can see that the solutions are values that are less than 1. All these values form the solution set to the inequality. Is there a way to write the solution set concisely, without using the number line and without writing out all the numbers less than 1?

Does the solution set have anything to do with the solution to the equation 7(3-x)=14?

Why does the solution set to the inequality 7(3-x)>14 involve numbers less than 1?

Which orchard would be cheaper to visit if the class has 8 students?

Which orchard would be cheaper to visit if the class has 12 students?

Which orchard would be cheaper to visit if the class has 30 students?

What does n represent in each statement?

In this situation, what does the equation 9(n+3)=10(n+1) mean?

What does the solution to the inequality 9(n+3)<10(n+1) tell us?

The equation 7.50h+\frac{50}{4}h=18h+33 represents a possible constraint about a situation. Solve the equation and check your solution.

Here is a graph on a number line. Put a scale on the number line so that the point marked with a circle represents the solution to the equation.

Does one job pay better if the student works fewer hours than the solution you found earlier? If so, which job?

Does one job pay better if the student works more hours than the solution you found earlier? If so, which job?

Here are two inequalities and two graphs that represent the solutions to the inequalities. Match each inequality with a graph that shows its solution.

Solve this equation and check your solution: -\frac{4(x+3)}{5}=4x-12

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose a couple of values less than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose a couple of values greater than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose 2 for x. Is it a solution?

Graph the solution to the inequality.

How does solving the equation 4x-3=12(x+3) help with solving the inequality 4x-3\geq12(x+3)

Which graph correctly shows the solution to the inequality \frac{7x-3}{9}\geq8-2x? Show or explain your reasoning.

Lesson 19: Solutions to Inequalities in One Variable

Rewrite the student learning goals in your own words.

Write some solutions to the inequality y\leq9.2. Be prepared to explain what makes a value a solution to this inequality.

Write one solution to the inequality 7(3-x)>14. Explain your reasoning.

How did you know that the value you chose is a solution?

What do you notice about all the points that are on the line?

On the number line, we can see that the solutions are values that are less than 1. All these values form the solution set to the inequality. Is there a way to write the solution set concisely, without using the number line and without writing out all the numbers less than 1?

Does the solution set have anything to do with the solution to the equation 7(3-x)=14?

Why does the solution set to the inequality 7(3-x)>14 involve numbers less than 1?

Which orchard would be cheaper to visit if the class has 8 students?

Which orchard would be cheaper to visit if the class has 12 students?

Which orchard would be cheaper to visit if the class has 30 students?

What does n represent in each statement?

In this situation, what does the equation 9(n+3)=10(n+1) mean?

What does the solution to the inequality 9(n+3)<10(n+1) tell us?

The equation 7.50h+\frac{50}{4}h=18h+33 represents a possible constraint about a situation. Solve the equation and check your solution.

Here is a graph on a number line. Put a scale on the number line so that the point marked with a circle represents the solution to the equation.

Does one job pay better if the student works fewer hours than the solution you found earlier? If so, which job?

Does one job pay better if the student works more hours than the solution you found earlier? If so, which job?

Here are two inequalities and two graphs that represent the solutions to the inequalities. Match each inequality with a graph that shows its solution.

Solve this equation and check your solution: -\frac{4(x+3)}{5}=4x-12

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose a couple of values less than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose a couple of values greater than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12Choose 2 for x. Is it a solution?

Graph the solution to the inequality.

How does solving the equation 4x-3=12(x+3) help with solving the inequality 4x-3\geq12(x+3)

Which graph correctly shows the solution to the inequality \frac{7x-3}{9}\geq8-2x? Show or explain your reasoning.

The equation 7.50h+\frac{50}{4}h=18h+33 represents a possible constraint about a situation.

Solve the equation and check your solution.

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values less than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values greater than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose 2 for x. Is it a solution?

The equation 7.50h+\frac{50}{4}h=18h+33 represents a possible constraint about a situation.

Solve the equation and check your solution.

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values less than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose a couple of values greater than 2 for x. Are they solutions to the inequality?

Consider the inequality: -\frac{4(x+3)}{5} \geq 4x-12

Choose 2 for x. Is it a solution?