Grade 7 · Unit 6 · Section C
This page follows the inequalities section of the guide, rewritten for students. You will move from reading simple inequalities, to solving them with boundary points, to modeling real situations where the answer has to make sense.
Open circle: the boundary is not included, as in x > 1.
Closed circle: the boundary is included, as in h ≥ 16.
The number line shows values of x that make the inequality x > 1 true.
A sign next to a roller coaster at an amusement park says, "You must be at least 60 inches tall to ride." Noah is happy to know that he is tall enough to ride.
The table shows four inequalities and four possible values for x. Decide whether each value makes each inequality true, and complete the table with "true" or "false."
We use inequalities to describe a range of numbers. In many places, you are allowed to get a driver's license when you are at least 16 years old. When checking if someone is old enough to get a license, we want to know if their age is at least 16. If h is the age of a person, then we can check if they are allowed to get a driver's license by checking if their age makes the inequality h > 16 (they are older than 16) or the equation h = 16 (they are 16) true.
Explain your reasoning before comparing with a classmate.
For each inequality, find two values for x that make the inequality true and two values that make it false.
Sample answers: for x + 3 > 70, true x = 70, 100 and false x = 0, -10. For x + 3 < 70, true x = 60, 0 and false x = 70, 100. For -5x < 2, true x = 1, 2 and false x = -1, -2. For 5x < 2, true x = 0, -1 and false x = 1, 100.
Here is an inequality: -3x > 18.
The values for -3x > 18 are any values less than -6. The values for -3x ≥ 18 are the same, except -6 is also included.
Solutions to Equations and Solutions to Inequalities
Andre has a summer job selling magazine subscriptions. He earns $25 per week plus $3 for every subscription he sells. Andre hopes to make at least enough money this week to buy a new pair of soccer cleats.
Kiran has $100 saved in a bank account. (The account doesn't earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least $25 in the account a year from now.
Jada and Diego baked a large batch of cookies.
The solution to 5 - 3x > 35 is either x > -10 or -10 > x. Which solution is correct? Explain how you know.
x < -10. For example, x = -100 gives 305 > 35, which is true.
The school band director determined from past experience that if they charge t dollars for a ticket to the concert, they can expect attendance of 1000 - 50t. The director used this model to figure out that the ticket price needs to be $8 or greater in order for at least 600 to attend. Do you agree with this claim? Why or why not?
No. t = 8 is the boundary for 1000 - 50t = 600, but higher prices lower attendance. The inequality should be 1000 - 50t ≥ 600, so t ≤ 8.
Draw the solution set for each of the following inequalities.
x ≤ 5 has a closed circle at 5 and shading left. x < 5/2 has an open circle at 2.5 and shading left.
Here is an inequality: -x ≥ -4.
Let's investigate the inequality x - 3 > -2.
Let's investigate -4x + 5 ≥ 25.
Here is an inequality: 3(10 - 2x) < 18. The solution to this inequality is all the values you could use in place of x to make the inequality true.
Consider the inequality -1 ≤ x/2.
For -1 ≤ x/2, the solutions are x ≥ -2. For 1 ≤ -x/2, the solutions are x ≤ -2.
Diego is solving the inequality 100 - 3x ≥ -50. He solves the equation 100 - 3x = -50 and gets x = 50. What is the solution to the inequality?
B. The solution is x ≤ 50. Testing x = 0 gives 100 ≥ -50, which is true.
Solve the inequality -5(x - 1) > -40, and graph the solution on a number line.
Open circle at 9 with shading left.
Select all values of x that make the inequality -x + 6 ≥ 10 true.
-4.01, -4, and -7.
For each inequality, find the value or values of x that make it true.
Choose the inequality that best matches each given situation. Explain your reasoning.
Choose one of the club situations from the last task. Explain how the inequality models the situation.
{3, 4, 5, 6} is a set of four consecutive integers whose sum is 18.
Explain your reasoning before comparing with a classmate.
Priya looks at the inequality 12 - x > 5 and says "I subtract a number from 12 and want a result that is bigger than 5. That means that the solutions should be values of x that are smaller than something."
Yes. When x = 7 the inequality is no longer true, but anything less than 7 is still true. The solution is x < 7.
When a store had sold 2/5 of the shirts that were on display, they brought out another 30 from the stockroom. The store likes to keep at least 150 shirts on display. The manager wrote the inequality 3/5x + 30 ≥ 150 to describe the situation.
The 3/5 means the fraction of the original shirts still on display after 2/5 were sold. The inequality gives x ≥ 200, so there must have been at least 200 shirts on display originally.
You know x is a number less than 4. Select all the inequalities that must be true.
B, C, and E.
Here is an unbalanced hanger.
Each triangle weighs more than 4 grams. Each circle weighs less than 4.5 grams.
Match each sentence with the inequality that could represent the situation.
A matches x + 2 < 20. B matches x - 2 < 20. C matches 2x < 20. D matches (1/2)x < 20.
The stage manager of the school musical is trying to figure out how many sandwiches he can order with the $83 he collected from the cast and crew. Sandwiches cost $5.99 each, so he lets x represent the number of sandwiches he will order and writes 5.99x ≤ 83. He solves this to 2 decimal places, getting x ≤ 13.86.
A mover is loading an elevator with many identical 48-pound boxes. The mover weighs 185 pounds. The elevator can carry at most 2000 pounds.
In a day care group, nine babies are five months old and 12 babies are seven months old.
We can represent and solve many real-world problems with inequalities. Whenever we write an inequality, it is important to decide what quantity we are representing with a variable. After we make that decision, we can connect the quantities in the situation to write an expression, and finally, the whole inequality.
28 students travel on a field trip. They bring a van that can seat 12 students. Elena and Kiran's teacher asks other adults to drive cars that seat 3 children each to transport the rest of the students.
Yes. n counts cars, so only whole numbers make sense. 12 + 3n ≥ 28 gives n ≥ 5 1/3, so 6 cars are needed; using > gives the same whole-number answer.
Answer each question.
Tables: 4x + 10 ≥ 200. Barrels: 4x + 10 ≥ 200. Both solve to x ≥ 47.5, but tables require whole numbers, so at least 48 tables; barrel capacity can be 47.5 liters or more.