Lesson 18Solutions to Inequalities in One Variable
Learning Goal
Let’s find and interpret 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.
Warm Up: Find a Value, Any Value
Problem 1
Write some solutions to the inequality
Problem 2
Write one solution to the inequality
Activity 1: 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.
Problem 1
Which orchard would be cheaper to visit if the class has:
8 students?
12 students?
30 students?
Problem 2
To help her compare the cost of her two options, the teacher first writes the equation
What does
represent in each statement? n this situation, what does the equation
mean? What does the solution to the inequality
tell us? Graph the solution to the inequality on the number line. Be prepared to show or explain your reasoning.
Activity 2: 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.
Problem 1
The equation
Solve the equation and check your solution.
Given the 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.
Problem 2
Does one job pay better if:
The student works fewer hours than the solution you found earlier? If so, which job?
The student works more hours than the solution you found earlier? If so, which job?
Problem 3
Here are two inequalities and two graphs that represent the solutions to the inequalities.
Inequality 1:
Inequality 2:
Put the same scale on each number line so that the circle represents the number of hours that you found earlier.
Write each inequality with the graph that shows its solution. Be prepared to explain or show how you know.
Graph A:
Graph B:
Activity 3: Equality and Inequality
Problem 1
Solve this equation and check your solution:
Problem 2
Consider the inequality:
Choose a couple of values less than 2 for
. Are they solutions to the inequality? Choose a couple of values greater than 2 for
. Are they solutions to the inequality? Choose 2 for
. Is it a solution? Graph the solution to the inequality on the number line.
Are you ready for more?
Problem 1
Here is a different type of inequality:
Is 1 a solution to the inequality? Is 3 a solution? How about -3?
Describe all solutions to this inequality. (If you like, you can graph the solutions on a number line.)
Describe all solutions to the inequality
. Test several numbers to make sure your answer is correct.
Lesson Summary
The equation
The inequality
One way to find the solutions to an inequality is by reasoning. For example, to find the solution to
Another way to find the solutions to
To find out the range of values that make the inequality true, we can try values less than and greater than 4 in our inequality and see which ones make a true statement.
Let’s try some values less than 4:
If
, the inequality is or , which is true. If
, the inequality is or , which is also true.
Let’s try values greater than 4:
If
, the inequality is or , which is false. If
, the inequality is or , which is also false.
In general, the inequality is false when
We can represent the solution set to an inequality by writing an inequality,