Inequalities and Multiplication

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Lesson 3-6 Inequalities and Multiplication BIG IDEA Multipling each side of an inequalit b a positive number keeps the direction of the inequalit; multipling each side b a negative number reverses

Vocabular inequalit boundar point interval endpoint > 3 > 3 Graphs and Inequalities on a Number Line An inequalit is a mathematical sentence with one of the verbs < ( is less than ), > ( is greater

The equation = 3 has just one solution, while the inequalities > 3 and 3 have infinitel man solutions. Their graphs at the right show these differences.

1 Lesson 3-6 Inequalities and Multiplication BIG IDEA Multipling each side of an inequalit b a positive number keeps the direction of the inequalit; multipling each side b a negative number reverses the direction of the inequalit. Vocabular inequalit boundar point interval endpoint > 3 > 3 Graphs and Inequalities on a Number Line An inequalit is a mathematical sentence with one of the verbs < ( is less than ), > ( is greater than ), ( is less than or equal to ), or ( is greater than or equal to ). Even though the look similar, equations and inequalities are different in important was. Consider = 3, > 3, and 3. The equation = 3 has just one solution, while the inequalities > 3 and 3 have infinitel man solutions. Their graphs at the right show these differences. In the graph of > 3 on a number line, the 3 is marked with an open circle because 3 does not make the sentence > 3 true. The number 3 is the boundar point between the values that satisf > 3 and those that do not. Numbers just a little larger than 3 such as 3.1 and 34 3 are solutions, as are larger numbers like 1 million. The graph 1, of 3 does include 3 because the sentence 3 3 is true. Another wa to write 3 is to use set-builder notation. It is written as {: 3} and is read as the set of all such that is greater than or equal to 3. When solving real-world problems, ou must often decide what domain makes sense for the situation. Inequalities are common in the world. For eample, let = a Fahrenheit temperature at which water is in solid form (ice). Then, since water freezes below 32ºF, < 32, as graphed at the right. The open circle at 32 shows that all numbers to the left of 32 are graphed, but not 32 itself. Infinitel Man Solutions Mental Math Estimate to the nearest dollar. a. 2% tip on a $49.6 bill b. 2% tip on a $149.6 bill c. 2% tip on a $249.6 bill Eactl One Solution Inequalities and Multiplication 1

Chapter 3 QY Some situations lead to double inequalities. Let E = the elevation of a place in the United States.

This double inequalit is graphed below. QY Write the solutions to the inequalit < 32 in setbuilder notation.

An interval is a set of numbers between two numbers a and b, which are called the endpoints of the interval. When the endpoints are included in the interval, it is described b a b.

2 Chapter 3 QY Some situations lead to double inequalities. Let E = the elevation of a place in the United States. Elevations in the United States range from 86 meters below sea level (in Death Valle) to 6,194 meters above sea level (at the top of Denali, also known as Mt. McKinle). So, 86 6,194. This double inequalit is graphed below. QY Write the solutions to the inequalit < 32 in setbuilder notation. 86 6,194 1, 1, 2, 3, 4,, 6, 7, Draw dots on 86 and 6,194 to show that those numbers are included in the solution. The graph of 86 6,194 is an interval. An interval is a set of numbers between two numbers a and b, which are called the endpoints of the interval. When the endpoints are included in the interval, it is described b a b. If the endpoints are not included, the is replaced b <, giving a < < b. The height of Mt. McKinle (Denali) is 2,32 feet. Source: U.S. Geological Surve Activit The inequalities in 1 7 below are ver similar. Use a CAS to solve each one. Record the operation ou do to both sides and the inequalit that results. A CAS screen for the fi rst problem is shown < < < > In 8 and 9, give what ou epect the solution to be. Use a CAS to check our answer. 8. 4m > w The inequalities ou solved in 1 9 can be grouped into two categories whose solution processes are somewhat different. What are the two categories? The Multiplication Propert of Inequalit Here are some numbers in increasing order. Because the numbers are in order, ou can put the inequalit sign (<) between an two of them. 1 < 6 < < < 1 16 Linear Equations and Inequalities

3 Now multipl these numbers b some fied positive number, sa The order stas the same, as shown in the number line at the right You could still put a < sign between an two of the numbers. This illustrates that if <, then 2 < 2. In general, multiplication b a positive number maintains the order of a pair or a list of numbers Multiplication Propert of Inequalit (Part 1) If < and a is positive, then a < a. Here is the same list of numbers we used earlier. 1 < 6 < < < 1 Now we multipl these numbers b 2 and something different happens Notice that the numbers in the original list are in increasing order, while the numbers in the second list are in decreasing order. The order has been reversed If ou multipl both sides of an inequalit b a negative number, ou must change the direction of the inequalit. This idea can be generalized. Multiplication Propert of Inequalit (Part 2) If < and a is negative, then a > a. Changing from < to >, or from to, or vice-versa, is called changing the sense of the inequalit. You have to change the sense of an inequalit when ou are multipling both sides b a negative number. Otherwise, solving a < b is similar to solving a = b. Solving Inequalities To solve an inequalit of the form a < b, ou must isolate the variable on one side ( just like solving an equation). Be careful to notice whether ou multipl or divide each side b a positive number or a negative number. Inequalities and Multiplication 17

Chapter 3 Eample 1 Solve 7 126 and check. Solution Multipl both sides b 7, the reciprocal of 7.

Check 126 7 18 Simplif. Simplif. Step 1 Substitute 18 for. Does 7 18 = 126? Yes. READING MATH Inequalities with variables are open sentences.

4 Chapter 3 Eample 1 Solve and check. Solution Multipl both sides b 7, the reciprocal of 7. Since 7 is a negative number, Part 2 of the Multiplication Propert of Inequalit tells ou to change the sense of the inequalit sign from to Write the inequalit Multipl each side b 7. Check Simplif. Simplif. Step 1 Substitute 18 for. Does 7 18 = 126? Yes. READING MATH Inequalities with variables are open sentences. When the variable is replaced with a number, the inequalit is either true or false. Step 2 Tr a number satisfing < 18. We use 2. Is ? Yes, The two-step check of an inequalit is important. The first step checks the boundar point in the solution. The second step checks the sense of the inequalit. GUIDED Eample 2 a. Solve 2 4. b. Graph the solution. c. Check our answer. Solutions a. 2 4 Write the inequalit. 4 2? 4 4 Multipl each side b 4.? Simplif. This can be rewritten as?. b. Graph the solution on the number line c. Step 1 Check the boundar point b substituting for. Does? = 4(? )? 18 Linear Equations and Inequalities

Step 2 Check whether the sense of the inequalit is correct. Pick some number from the shaded region in Part b. This number should also work in the original inequalit. We choose. Is? 4(? )? Yes,?

5 Step 2 Check whether the sense of the inequalit is correct. Pick some number from the shaded region in Part b. This number should also work in the original inequalit. We choose. Is? 4(? )? Yes,??. Since both steps worked, the solution to 2 4 can be described b the sentence?. Eamples 1 and 2 showed solutions of two tpes of inequalities. Multiplication Propert of Inequalit (in words) You ma multipl both sides of an inequalit b the same positive number without affecting the set of solutions to the sentence. You can also multipl both sides b a negative number, but then ou must also change the sense of the inequalit. Questions COVERING THE IDEAS In 1 and 2, consider the situation. a. Write an inequalit that describes the situation. b. Identif the boundar point for the inequalit. c. Graph the solution set of the inequalit. 1. To successfull leave Earth s orbit, a satellite must be launched at a velocit of at least 11.2 kilometers per second. Let v be the launch velocit of a satellite in kilometers per second. 2. To obtain a passing grade on the Spanish eam, Marlene needs to answer at least 6 percent of the test items correctl. Let p be the percentage of items answered correctl. 3. Wakana s dog w weighs over 12 pounds, but not more than 13 pounds. Graph the possible values of w on a number line. In 4 and, graph the inequalit on a number line. 4. < Consider the inequalit 2 < 3. What true inequalit results if ou multipl both sides of the inequalit b the given number? a. 6 b. c. 4 2 In 7 12, solve the inequalit. Then check our answer > < 2n z a b 6 2 The SBS-4 communications satellite is deploed from the space shuttle paload ba. Source: National Aeronautics and Space Administration Inequalities and Multiplication 19

6 Chapter Consider the following list of numbers. 2, 3, 4,, 6 a. Place an appropriate smbol between each pair of consecutive numbers (<,,, or >). b. Multipl each number in the list b 8, then place an appropriate smbol between each pair of consecutive numbers (<,,, or >) in the new list. c. Did the direction of the smbols change in Part b? Wh or wh not? APPLYING THE MATHEMATICS 14. Consider the following list of numbers. 1, 2, 3, 4, a. Place an appropriate smbol between each pair of consecutive numbers (<,,, or >). b. Create a new list b taking the reciprocal of each number, then place an appropriate smbol between each pair of consecutive numbers (<,,, or >) in the new list. c. Did the direction of the smbols change in Part b? Wh or wh not? 1. Fill in the Blanks If 6 m 6.2, then? m?. In 16 and 17, read the problem situation. a. Write an inequalit describing the situation. b. Solve the inequalit. 16. An engineer is designing a rectangular parking lot that is to be 7 feet wide. According to the cit building code, the area of the lot can be at most 6,2 square feet. What is the allowable length of the lot? 17. A concert hall is being designed to seat at least 2,2 people. Each row will have 8 seats. How man rows of seats will the hall have? REVIEW 18. The boes are of unknown equal weight W. (Lesson 3-) oz oz oz oz W W oz oz oz oz oz oz oz oz oz oz oz oz a. What equation is pictured b the balance above? b. What two steps can be done with the weights on the balance to find the weight of a single bo? c. How much does each bo weigh? 16 Linear Equations and Inequalities

7 In 19 and 2, solve the equation. (Lesson 3-) 19. 4(3-1.) + 7 = (a + ) - 3( + a) = Grafton went to the store to bu bottles of soda and bags of chips for a part. He bought bottles of soda for $1.99 each and bags of chips for $2.99 each. He bought twice as man bags of chips as bottles of soda. After paing with two twent-dollar bills, he received $.1 in change. (Lessons 3-, 3-3) a. Define a variable and write an equation describing the situation. b. How man bottles of soda and bags of chips did Grafton bu? 22. Multiple Choice How do the solutions to = 3 and 3 = compare? (Lessons 3-4, 2-8, 2-4) A The are equal. C The are reciprocals. B The are opposites. D none of the above In 23 and 24, a fact triangle is given. Write the related facts and determine the value of. (Lesson 2-7) ( 1) Tomás drove at an average speed of 61 miles per hour for 3 hours. About how man miles did he travel? (Previous Course) 4 EXPLORATION 26. A rectangle is 12 units b w units. a. Find the values of w that would make the area of the rectangle greater than 84 square units. b. Find the values of w that would make the area of the rectangle less than or equal to 216 square units. c. Write a sentence to eplain what the inequalit 6 12w < 18 means in relation to the rectangle. d. Solve the inequalit in Part c and eplain its meaning. 12 w QY ANSWER {: < 32} Inequalities and Multiplication 161

LESSON 4.3 GRAPHING INEQUALITIES

LESSON 4.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES LESSON.3 GRAPHING INEQUALITIES 9 OVERVIEW Here s what ou ll learn in this lesson: Linear Inequalities a. Ordered pairs as solutions of linear inequalities b. Graphing linear