Warm-Up By Jacob B A guy came up to me and offered me a chocolate bar for the price of the product of a number and seven, increased by nine, then decreased by six, then increased be three squared. Which equaled nineteen. How much does the chocolate bar cost?

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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

Additional Example 1: Completing an Inequality Compare. Write. A. 23 – > B. 5(12) <

Try This: Example 1 Compare. Write. A. 19 – < B. 4(15) >

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.

x < 5 4 < 5 x = < 5 x is less than 5 Word Phrase Inequality Sample Solutions Solution Set x = 4

a > 0 7 > 0 a = 2525 > 0 a is greater than 0 a is more than 0 Word Phrase Inequality Sample Solutions Solution Set –3 –2 – a = 7

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

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

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.

Additional Example 2A: Solving and Graphing Inequalities Solve and graph the inequality. A. x  8 –2.5 x  Subtract 2.5 from both sides. According to the graph, 5.4 is a solution, since

Additional Example 2B: Solving and Graphing Inequalities Solve and graph the inequality. B. 5t > t > t > 15 Divide both sides by 5.

Additional Example 2C: Solving and Graphing Inequalities Solve and graph the inequality. C. w – 1 < 8 w < 9 – Add 1 to both sides.

Additional Example 2D: Solving and Graphing Inequalities Solve and graph the inequality. D. 3  12  p Multiply both sides by 4. p 4 3  p 4 4 4

Try This: Examples 2A and 2B Solve and graph each inequality. A. x + 2  3.5 –2 x  Subtract 2 from both sides. B. 6u > u > u > 72 Divide both sides by 6.

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

Additional 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?

Additional 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  88 feet

Additional 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 feet

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

Additional Example 3 Continued Look Back 4 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 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 20x  130 Solve 3 20 x  6.5 At least 6.5 packages of cookies need to be purchased. Divide both sides by 20.

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

Lesson Quiz Use to compare each inequality (2) – 2 11 Solve and graph each inequality. 3. k + 9 <  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. > 6  m > k< 3 m 2 –5 –4–3–2– –4 –3–2– (21)  64; 63  64; yes ?