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

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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.
1

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.
1

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

1

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

1

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

1

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

1

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?

1

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

1

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.
1

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

1

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

1

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).
1

What does n represent in each statement?

1

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

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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.
1

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

Solve the equation and check your solution.

1

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.

1

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

1

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

1

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 itemCorresponding Item
7.50h+\frac{50}{4}h<18h+33
7.50h+\frac{50}{4}h>18h+33
19.4 Equality and Inequality
1

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

1

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?

1

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?

1

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

Choose 2 for x. Is it a solution?

1

Graph the solution to the inequality.

Lesson Synthesis
1

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

19.6 Seeking Solutions
1

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