lesson 12.2.2 version 2.5-studentnebula2.deanza.edu/~mo/Math217/05-lesson_12.2.2... · statway®"studenthandout""|3""" lesson"12.2.2"" solvinginequalities!! ©2013!thecarnegie!foundation!for!theadvancement!ofteaching!

© 2013 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 2.5, STATWAY® -‐ STUDENT HANDOUT

STATWAY® STUDENT HANDOUT

Lesson 12.2.2 Solving Inequalities

INTRODUCTION Example I Mr. Lawson bought a car for $16,000. Each year the car loses some value. If the worth W of the car x years after he bought it is modeled by ! = −1500! + 16000, when will the car be worth less than $11,500? The graph of the model is below.

Worth of Car

The horizontal line (! = 11,500) represents our “threshold.” We want to know when the value of the car will be below $11,500. 1 Use the graph to answer the following.

A In how many years after the purchase will the car be worth $11,500?

B During which years will the car be worth less than $11,500? Explain your reasoning.

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Example II Ms. Nizami loaned $10,000 to a start-‐up company. The terms of the loan say that the amount of money, A, that the company owes at the end of x months will be ! = 10000 + 120!. When will the company owe less than $10,600 on the loan? The graph of the model is below.

Total Owed on the Loan

2 Use the graph to answer the following questions.

A How many months will it take until the company owes $10,600?

B During which months is the amount of money that the company owes less than $10,600? Explain your reasoning.

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NEXT STEPS Understanding Direction 3 To help you visualize the relationships between the numbers, label the number line below with numbers

from -‐7 to 7, in increments of 1.

If a is a number to the left of b on the number line, then a < b. Since 2 is to the left of 5 on the number line above, we know that 2 < 5. Perform the indicated operations on the original inequalities below. Be careful, because some inequalities may change!

Original Inequality Multiply by 4 Multiply by -‐3 Divide by 2 Divide by -‐2

4 < 6 16 < 24

–2 < 4 –8 < 16

–6 < –2

Summarize the pattern that you see: 4 When an inequality is multiplied by a positive number, its direction is __________________. 5 When an inequality is multiplied by a negative number, its direction is __________________. 6 When an inequality is divided by a positive number, its direction is __________________. 7 When an inequality is divided by a negative number, its direction is __________________.




Solving Inequalities Just like equations, inequalities can have variables, and we can solve them. Solving inequalities is similar to solving equations – we can add, subtract, multiply, and divide both sides by any number (besides zero). Be careful though! If you multiply or divide by a negative, the inequality changes direction. 8 Solve the inequality: 4! < 12. 9 The solution to question 8 is actually a collection of many, many numbers. Shade the region that

represents the solution to question 8 on the number line below. If the endpoint is included, it should be shown with a “closed” dot. If the starting number is not included it should be shown with an open dot.

TRY THESE 10 Solve, and shade the region that represents the solution on the number line: −5! < 30.

11 Solve the inequality and graph the solution: −3! + 7 < 46. Check your answer at ! = 0.

12 Solve the inequality and graph the solution: 5! + 7 < 8! − 11. Check your answer at ! = 0.




We now look back at Examples I and II from earlier in this lesson. This time, we want to solve the inequalities using an algebraic process. 13 Refer to Example I. Mr. Lawson bought a car for $16,000. Each year the car loses some value.

The car’s worth is represented by W. The number of years after Mr. Lawson bought the car is represented by x. The equation that gives the car’s worth after x years is ! = 16000 − 1500!. When will the car be worth less than $11,500?

A What inequality could be used to answer this question?

B Solve the inequality. Check to make sure that your answer matches your answer from 1B.

14 Refer to Example II. Ms. Nizami loaned $10,000 to a start-‐up company. The terms of the loan read

that the amount of money, A, the company owes at the end of x months will be ! = 10000 +120!. When is the amount of money that the company owes less than $10,600?

A What inequality could be used to answer this question?

B Solve the inequality. Check to make sure that your answer matches 2B.

NEXT STEPS Solving Inequalities with a Graph We have seen that solving inequalities is similar to solving equations. There are times when solving inequalities is not like solving equations. One important exception occurs when we multiply or divide by a negative number. When we multiply or divide by a negative number, we have to reverse the direction of the inequality.




Another way to solve an inequality is with a graph. One way to solve inequalities with graphs is given here.

(1) Write a formula for each side of the inequality.

(2) Graph each of those formulas.

(3) Extend the graphs as needed to find a point or points where they cross (or make it clear that they don’t ever cross.)

(4) Use the graph or a table to estimate the point(s) at which they cross OR write and solve an equation to find the crossing points.

(5) Use the graph to see the interval(s) for which the inequality holds, and express this in words.

TRY THIS A Quadratic Inequality Someone throws a ball straight upward with a velocity of 128 feet per second. They throw the ball from the top of a tower which is 144 feet high. Let x = number of seconds since the ball was thrown and y = height of the ball at time x. The model for the height is ! = 144 + 128! − 16!!. When is the height of the ball greater than 336 feet? We will solve this inequality: ! > 336 The graph of ! = 144 + 128! − 16!! is shown below.

Height of the Ball

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15 Let’s find the solution using the graph.

A How many seconds after it is thrown from the tower will it be before the ball passes 336 feet? B Does the ball pass a height 336 feet high more than once? C When is the ball more than 336 feet above the ground?




TAKE IT HOME 1 Solve the following inequalities and check your answer. Graph the solution on the number line.

A 8! > 4

B 63 > −7!

C 8 − 5! > 43

D 3 + 2! < 6! − 21

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Ball Height

2 The world record time (in minutes) for the 1500 meter run for years in the 1900’s can be modeled by the formula ! = 4 − 0.006!. In this formula, t is the number of years after 1900 and R is the world record time the model predicts for that year. In how many years, from the year 1900, does this model predict that the world record will be less than 3.5 minutes?

3 Two measurements for temperature are Celsius and Fahrenheit. In the United States we use

Fahrenheit to measure temperature. The formula for converting temperature Celsius to Fahrenheit is ! = 1.8! + 32. One medical website says that if a baby has a temperature of at least 100.4°F, the baby should be seen by a doctor. In many countries, however, Celsius is the measurement for temperature. According to this medical advice website, at what temperatures C should the baby be examined by a doctor?

4 An organization can purchase a bulk-‐mail permit for $190 per year. With the bulk mail permit,

each piece of mail costs 27.6 cents to send. Without the permit, each piece of mail costs 44 cents to send. How many pieces of mail would an organization need to send in a year to make it cheaper to use bulk mail?

5 Optional: A person standing on a tower that is 125 ft tall

throws a ball straight up with a velocity of 160 ft/sec. The mathematical model for the height, y, of the ball at x seconds after it is thrown is

! = −16 ! − 5 ! + 525. When is this ball at least 381 ft high? In the graph, a horizontal line is drawn at 381 ft.




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This lesson is part of STATWAY®, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the Statway® Networked Improvement Community, please visit carnegiefoundation.org.

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lesson 12.2.2 version 2.5-studentnebula2.deanza.edu/~mo/Math217/05-lesson_12.2.2... · statway®"studenthandout""|3""" lesson"12.2.2"" solvinginequalities!! ©2013!thecarnegie!foundation!for!theadvancement!ofteaching!