Objective: solve and graph inequalities Chapter 1 Algebra Toolbox 1-5 Solving Simple Inequalities Objective: solve and graph inequalities

Agenda Warm Up Problem of the Day Lesson Presentation Practice Activity Quiz

Warm Up Solve. 1. x + 6 = 13 2. 8n = 48 3. t  2 = 56 x = 7 4. 6 = z 6 z = 36

Problem of the Day Bill and Brad are taking drivers education. Bill drives with his instructor for one and a half hours three times a week. He needs a total of 27 hours. Brad drives two times a week, two hours each time. He needs 26 hours. Who will finish his hours first? Bill

Learn to solve and graph inequalities.

Vocabulary inequality algebraic inequality solution of an inequality solution set

An inequality compares two quantities and typically uses one of these symbols:  < is less than is greater than   is less than or equal to is greater than or equal to

Example 1: Completing an Inequality Compare. Write < or >. A. 23 – 14 6 9 6 > B. 5(12) 70 60 70 <

Try This: Example 1 Compare. Write < or >. A. 19 – 3 17 16 17 < B. 4(15) 50 60 50 >

An inequality that contains a variable is an algebraic inequality. A number that makes an inequality true is a solution of the inequality. The set of all solutions is called the solution set. The solution set can be shown by graphing it on a number line.

Word Phrase Sample Solutions Inequality Sample Solutions Solution Set x is less than 5 x < 5 x = 4 4 < 5 x = 2.1 2.1 < 5 1 2 3 4 5 6 7

Word Phrase Sample Solutions Inequality Sample Solutions Solution Set a is greater than 0 a is more than 0 a > 0 a = 7 7 > 0 a = 25 25 > 0 –3 –2 –1 0 1 2 3

Word Phrase Sample Solutions Inequality Sample Solutions Solution Set y is less than or equal to 2 y is at most 2 y  2 y = 0 0  2 y = 1.5 1.5  2 –3 –2 –1 0 1 2 3

Word Phrase Sample Solutions Inequality Sample Solutions Solution Set m is greater than or equal to 3 m is at least 3 m  3 m = 17 17  3 m = 3 3  3 –1 0 1 2 3 4 5

Most inequalities can be solved the same way equations are solved. Use inverse operations on both sides of the inequality to isolate the variable. There are special rules when multiplying or dividing by a negative number, which you will learn in the next chapter.

Example 2A: Solving and Graphing Inequalities Solve and graph the inequality. A. x + 2.5  8 –2.5 –2.5 Subtract 2.5 from both sides. x  5.5 1 2 3 4 5 6 7 According to the graph, 5.4 is a solution, since 5.4 < 5.5, and 6 should not be solution because 6 > 5.5.

Example 2B: Solving and Graphing Inequalities Solve and graph the inequality. B. 5t > 15 5t > 15 Divide both sides by 5. 5 5 t > 3 1 2 3 4 5 6 7

Example 2C: Solving and Graphing Inequalities Solve and graph the inequality. C. w – 1 < 8 + 1 + 1 Add 1 to both sides. w < 9 –3 0 3 6 9 12 15

Example 2D: Solving and Graphing Inequalities Solve and graph the inequality. p 4 D. 3  3  p 4 4 • 4 • Multiply both sides by 4. 12  p 0 3 6 9 12 15 18

Try This: Examples 2A and 2B Solve and graph each inequality. A. x + 2  3.5 –2 –2 Subtract 2 from both sides. x  1.5 1 2 3 4 5 6 7 B. 6u > 72 6u > 72 Divide both sides by 6. 6 6 u > 12 3 6 9 12 15 18 21

Try This: Examples 2C and 2D Solve and graph each inequality. C. z – 6 < 15 + 6 + 6 Add 6 to both sides. z < 21 –21 –14 –7 0 7 14 21 D. 2  b 9 2  b 9 9 • 9 • Multiply both sides by 9. 18  b 0 3 6 9 12 15 18

Example 3: Problem Solving Application An interior designer is planning to place a wallpaper border along the edges of all four walls of a room. The total distance around the room is 88 feet. The border comes in packages of 16 feet. What is the least number of packages that must be purchased to be sure that there is enough border to complete the room?

Understand the Problem Example 3 Continued 1 Understand the Problem The answer will be the least number of packages of border needed to wallpaper a room. List the important information: The total distance around the room is 88 feet. The border comes in packages of 16 feet. Show the relationship of the information: the number of packages of border the length of one package of border Distance around the room • 

Example 3 Continued 2 Make a Plan •  Use the relationship to write an inequality. Let x represent the number of packages of border. x • 16 ft  88 ft

Example 3 Continued Solve 3 16x  88 16x  88 Divide both sides by 16. 16 16 x  5.5 At least 5.5 packages of border must be used to complete the room.

Example 3 Continued 4 Look Back Because whole packages of border must be purchased, at least 6 packages of border must be purchased to ensure that there is enough to complete the room.

Try This: Example 3 Ron will provide 130 cookies for the school fundraiser. He has to buy the cookies in packages of 20. What is the least number of packages Ron must buy to be sure to have enough cookies?

Understand the Problem Try This: Example 3 1 Understand the Problem The answer will be the number of packages of cookies a customer needs to purchase. List the important information: Cookies are sold in packages of 20 cookies. A customer needs to purchase 130 cookies. Show the relationship of the information: the number of packages of cookies to be purchased the number of cookies in one package 130 cookies • 

Try This: Example 3 Continued 2 Make a Plan Use the relationship to write an inequality. Let x represent the number of packages of cookies.  x • 20 cookies 130 cookies

Try This: Example 3 Continued Solve 3 20x  130 20x  130 Divide both sides by 20. 20 20 x  6.5 At least 6.5 packages of cookies need to be purchased.

Try This: Example 3 Continued 4 Look Back Because whole packages of cookies must be purchased, at least 7 packages of cookies must be purchased for the party.

Lesson Quiz Use < or > to compare each inequality. 1. 13 5(2) 2. 14 – 2 11 Solve and graph each inequality. 3. k + 9 < 12 4. 3  5. A school bus can hold 64 passengers. Three classes would like to use the bus for a field trip. Each class has 21 students. Write and solve an inequality to determine whether all three classes will fit on the bus. > > k< 3 –5 –4–3–2–1 0 1 2 3 4 5 m 2 6  m –4 –3–2–1 0 1 2 3 4 5 6 3(21)  64; 63  64; yes ?

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