~ 6.17 ~
Modeling with Inequalities
Learning Targets
I can use what I know about inequalities to solve real-world problems.
Notes
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.
As we are solving the inequality or equation to answer a question, it is important to keep the meaning of each quantity in mind. This helps us to decide if the final answer makes sense in the context of the situation.
For example: Han has 50 centimeters of wire and wants to make a square picture frame with a loop to hang it that uses 3 centimeters for the loop. This situation can be represented by 3 + 4s = 50, where s is the length of each side (if we want to use all the wire). We can also use 3 + 4s ≤ 50 if we want to allow for solutions that don’t use all the wire. In this case, any positive number that is less or equal to 11.75 cm is a solution to the inequality. Each solution represents a possible side length for the picture frame since Han can bend the wire at any point. In other situations, the variable may represent a quantity that increases by whole numbers, such as with numbers of magazines, loads of laundry, or students. In those cases, only whole-number solutions make sense.
Activities
17.1 Possible Values
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 represent the number of sandwiches he will order and writes 5.99x ≤ 83. He solves this to 2 decimal places, getting x ≤ 13.86.
Which of these are valid statements about this situation? (Select all that apply.)
He can call the sandwich shop and order exactly 13.86 sandwiches.
He can round up and order 14 sandwiches.
He can order 12 sandwiches.
He can order 9.5 sandwiches.
He can order 2 sandwiches.
He can order -4 sandwiches.
17.2 Elevator
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.
Write an inequality that says that the mover will not overload the elevator on a particular ride.
Solve your inequality and explain what the solution means.
Graph the solution to your inequality on a number line.
If the mover asked, “How many boxes can I load on this elevator at a time?” what would you tell them?
17.4 Movies on a Hard Drive
Elena is trying to figure out how many movies she can download to her hard drive. The hard drive is supposed to hold 500 gigabytes of data, but 58 gigabytes are already taken up by other files. Each movie is 8 gigabytes. Elena wrote the inequality 8x + 58 ≥ 500 and solved it to find the solution x ≥ 55.25.
Explain how you know Elena made a mistake based on her solution.
Fix Elena’s inequality and explain what each part of the inequality means.
Summary
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Assignment
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