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Unit 1 Performance Tasks
Inequalities
Key Vocabulary
inequalities
(desigualdad)
ESSENTIAL QUESTION
1.
How can you use inequalities to solve real-world problems?
EXAMPLE 1
Amy is having her birthday party at a roller skating rink. The rink charges
a fee of $50 plus $8 per person. If Amy wants to spend at most $170 for
the party at the rink, how many people can she invite to her party?
CAREERS IN MATH Mechanical Engineer A mechanical engineer
is testing how different springs stretch for a given force. Force is
measured in units called Newtons, abbreviated N. A spring stretches 2.5
millimeters for each 1-N force applied.
a. Write an expression for the stretch of this spring in millimeters for a
given force x in Newtons. Use your expression to find how much the
spring stretches for a 6-N force.
2.5x; 15 mm
Let p represent the number of people skating at the party.
b. Write an expression for the stretch of this spring in centimeters for
a given force x in Newtons. Use your expression to find how much
the spring stretches for a 6-N force. Does the spring stretch the same
length as in part a? Explain.
50 + 8p ≤ 170
8p ≤ 120
Subtract 50 from both sides.
8p ___
__
≤ 120
8
8
Divide both sides by 8.
0.25x; 1.5 cm; Yes, the spring stretches 1.5 cm which is the same
p ≤ 15
as 15 mm.
Up to 15 people can skate, so Amy can invite up to 14 people to her party.
c. The spring is 4 cm long when unloaded. Write and solve an equation
for the force needed to stretch the spring to a length of 7 cm.
Include a description of the variable.
EXAMPLE 2
Determine which, if any, of these values makes the inequality
-7x + 42 ≤ 28 true: x = -1, x = 2, x = 5.
−7(−1) + 42 ≤ 28
−7(2) + 42 ≤ 28
7 = 0.25x + 4 or 70 = 2.5x + 40 where x is the force in Newtons;
Substitute each value for x
in the inequality and evaluate
the expression to see if a true
inequality results.
−7(5) + 42 ≤ 28
x = 2 and x = 5
x = 12 N
The amount a spring stretches for a 1-N force is called the spring
constant, and depends on the spring’s construction and material.
The spring constant applies when the spring is compressed, which
reduces its length, as well as when it is stretched.
EXERCISES
1. Prudie needs $90 or more to be able to take her family out to dinner.
She has already saved $30 and wants to take her family out to eat in
4 days. (Lesson 2.2)
d. Write an expression for the 4-centimeter spring under a compressive
force c. Then find the length of the spring under a 6-N compressive force.
© Houghton Mifflin Harcourt Publishing Company
a. Suppose that Prudie earns the same each day. Write an inequality
to find how much she needs to earn each day.
-0.25c + 4 or -2.5c + 40 where x is a positive compressive force in
Newtons; 2.5 cm or 25 mm
30 + 4x ≥ 90
b. Suppose that Prudie earns $18 each day. Will she have enough
money to take her family to dinner in 4 days? Explain.
2. Yolanda saves money to buy a $200 cell phone. She has $70, and saves
$25 each week.
Yes, x = 18 is a solution of the inequality.
a. Write and solve an inequality to find out how many full weeks Yolanda
must save to have at least enough money to buy the cell phone.
Solve each inequality. Graph and check the solution. (Lesson 2.3)
70 + 25w > 200; w > 5.2; 6 full weeks or more
3. 7x - 2 ≤ 61
2. 11 - 5y < -19
b. Yolanda buys her phone when she has enough money saved. Will
she also have enough to buy a $35 protective case? Explain.
x≤9
y>6
© Houghton Mifflin Harcourt Publishing Company
MODULE
No; in 6 weeks, she has $70 + $25 × 6 = $220; $220 - $200 = $20
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Unit 1
$B0)/(6(B86*LQGG
12
61
30
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Unit 1
$B0)/(6(B86*LQGG
30
Expressions, Equations, and Inequalities
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