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INEQUALITIES ENRICHED

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> is the “greater than” symbol. It signifies that the value on the left hand side of the inequality is larger than the value on the right hand side of the inequality. ex: 9 > 3n is read < is the “less than” symbol. It signifies that the value on the left hand side of the inequality is smaller than the value on the right hand side of the inequality. ex: 4v < 19 is read How Do You Read an Inequality? “nine is greater than three times a number” “four times a number is less than nineteen”

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How Do You Read an Inequality? ≥ is the “greater than or equal to” symbol. It signifies that the value on the left hand side of the inequality is equal to or larger than the value on the right hand side of the inequality. ex: 3x ≥ 9is read ≤ is the “less than or equal to” symbol. It signifies that the value on the left hand side of the inequality is equal to or smaller than the value on the right hand side of the inequality. ex: 4x ≤ 24 is read “three times a number is greater than or equal to nine” “four times a number is less than or equal to twenty-four”

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How Do You Read an Inequality? Examples: a)14 > 2n ______________________________________ b)15 ≥ 3 + n ____________________________________ c)23 ≤ 30 – n ___________________________________ d)9 + n < 24 – n ________________________________ fourteen is greater than two times a number fifteen is greater than or equal to three plus a number twenty-three is less than or equal to thirty minus a number nine plus a number is less than twenty- four minus that same number.

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COMPOUND INEQUALITIES There are times when you will come across an inequality with two symbols in it. There are two forms an inequality of this type will take. TYPE 1: “AND” statements (arrows face the same direction) Ex: 3 < x < 9 is read… To read an inequality such as this, break it into smaller pieces. Compare the middle term to the term on the left and then to the term on the right. EXTENSION “a number is greater than threeand less than nine”

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COMPOUND INEQUALITIES There are times when you will come across an inequality with two symbols in it. There are two forms an inequality of this type will take. TYPE 2: “OR” statements (arrows face opposite directions) Ex: x 9 is read… This is easier to read since it is already broken up for us! EXTENSION “a number is less than 3 or greater than 9”

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COMPOUND INEQUALITIES a)2 < 5m ≤ 10: ________________________________ b)4 > 3n ≥ 0: _________________________________ c)r > 6 or r < 3: ________________________________ d)y ≤ 9 or y ≥ 13: ______________________________ EXTENSION Reading Compound Inequalities five times a number is greater than two and less than or equal to ten three times a number is less than four and greater than or equal to zero a number is greater than six or less than three a number is less than or equal to nine or greater than or equal to thirteen

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When given an inequality that contains a variable you can determine if a given solution is true or false by simply substituting the value in the inequality. ex 1: Is 9 + n > 22 when n = 10? ex 2: If b = 15 is + 8 ≥ 13 a true statement? Is It a Solution? No, if you plug in 10 for n then you get “19 is greater than 22” which is a false statement. Yes, if you plug in 15 for n then you get “13 is greater than or equal to 13” which is a true statement.

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Which of the following inequalities is/are incorrect when n = 10? a)6n + 8 > 68 b)5(6 + n) ≥ 80 c) – 10 < 0 d) ≤ 9 Is It a Solution? No, if you plug in 10 for n then you get “68 is greater than 68” which is a false statement. No, if you plug in 10 for n then you get “10 is less than or equal to 9” which is a false statement

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Which Symbol Should Be Used? Greater ThanLess Than Greater Than or Equal ToLess Than or Equal To a)Is more than b)Is greater than c)Is larger than d)Is above e)Is bigger than a)Is less than b)Is smaller than c)Is below a)Is at least b)Is no less than c)Is no smaller than d)Is at minimum a)Is at most b)Is no more than c)Is no greater than d)Is at maximum

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a)five is greater than a number: _____ b)sixteen is less than or equal to a number: ______________ c)two times the difference of a number and four is greater than or equal to seventy five: ______________ d) fourteen less than a number is at least seventeen: _____________ e)the difference of half a number and seven is no more than eighteen: ______________ Translating Inequalities 5 > n 16 ≤ n 2(n – 4) ≥ 75 n – 14 ≥ 17

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INEQUALITIES

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The lunch lady records the number of sandwiches sold in the school cafeteria each day. If she sells more than 50 peanut butter sandwiches, she orders more peanut butter. If s represents the number of sandwiches sold, write an inequality for this scenario.

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S > 50

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Mrs. Stanfield must have a minimum of 10 fish in her fish tank (at all times) in the front of the Intermediate Building. If f represents the number of fish, write an inequality to represent this scenario.

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F ≥ 10

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In NYC, the elevator ride to the top of the empire state building can hold no more than 3500 pounds. If p represents the number of pounds, write an inequality to represent this scenario.

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P ≤ 3500

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The gym class’s average results for girls participating in the long jump is 100 inches. Sue could jump no farther than 4 inches more than the average distance for girls. If j represents the distance Sue could jump, write an inequality to represent how far Sue jumped.

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j ≤ 8ft 8inches

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Aussie has a minimum of 5 pairs of UGGs boots in her closet at all times. If u represents the number of UGGs in her closet, write an inequality for this scenario.

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u ≥ 5

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Jimmy just bought a new refrigerator on BLACK FRIDAY. It has a special meat drawer that can hold no more than 10 lbs of meat. Write an inequality for this scenario.

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Let m = pounds of meat m ≤ 10

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George the elf has c candy canes. George got into a scuffle with his sister. He ended up breaking 2 of the candy canes. He needs a minimum of 25 unbroken candy canes for his Christmas party. Write an inequality for the scenario.

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Let c = unbroken candy canes c – 2 ≥ 25

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Julia has $20 to spend on ice cream. Each ice cream cone is $2.75. Julia buys ice cream for her friends, f. She wants to have a minimum of $10 left in her wallet after she buys all of her friends ice cream. Write an inequality for this scenario.

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20 – 2.75f ≥ 10

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Evergreen Farms are selling Christmas trees. They must sell at least 100 trees to stay in business this year. The first week they sold 30 trees. The second week they sold 60 trees. The next week they sell t trees. Write an inequality to represent this situation.

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Let c = trees sold 90+ c ≥ 100 Yes they will meet their goal

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If the Evergreen Farms sells 15 trees will they stay in business? Yes they will meet their goal and exceed it by 5 trees

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Carl is measuring a room in his home that he needs to purchase new carpeting for. A diagram of the room is shown below. The local carpeting store currently is offering a 20% discount if you purchase at least 120 sq. ft. of carpeting. If the formula for determining the area of a rectangle is A = bh write an inequality to represent the area of Carl’s room if he hopes to receive the discount. Is It a Solution? n 16 If n = 8, will Carl receive a discount? Explain Yes, Carl will need 128 sq. ft. which is more than the required 120 sq. ft. 16n ≥ 120

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Keith has $500 in his savings account at the beginning of June. He wants to have more than $200 in the account on December 1 st so that he can purchase holiday gifts for his family and friends. Keith withdraws $75 each month for his own expenses. a)Write an inequality to represent Keith’s situation. b)December is 6 months away… will he meet his goal? Explain. Is It a Solution? let m = months 500 – 75m > 200 No, when we substitute 6 in for m we have an incorrect inequality. It reads “50 is greater than 200” which is not true.

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Kelly is collecting canned foods to contribute to the Thanksgiving food drive at her Church. The box Kelly is placing the cans in can hold a maximum of 82.5 pounds. With a month until she has to turn in the canned goods Kelly has collected 53.25 pounds worth of canned goods. Kelly plans to collect as many more cans as she can but is not sure if she will need an additional box. a)Write an inequality to represent Kelly’s situation. b)If Kelly collects an additional 29.1 pounds of food will she need an additional box? Is It a Solution? let p = pounds 53.25 + p ≤ 82.5 No, when we substitute 29.1 in for p we find that “82.35 is less than or equal to 82.5” which is a true statement.

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So far this year Martina has earned test scores of 72%, 91%, 84%, and 82%. Martina would like her average to be at least an 85%. To earn this grade the sum of her tests must be a minimum of 425. a)Write an inequality that represents Martina’s situation. b)If Martina earns a 96% on her next test will she meet her goal? Explain. Is It a Solution? let m = Martina’s last test score 72 + 91 + 84 + 82 + m ≥ 425 Yes, when we substitute 96 in for m we have an inequality that reads “425 is greater than or equal to 425.” This is a true statement.

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You can use a number line to represent inequalities. Step 1: If you are not given a number line that is already numbered, draw a number line that contains the number given in the inequality, a number bigger than the number given, and a number smaller than the number given. Step 2: Place a dot on the number line For ≤ and ≥ use a closed circle on the number line o For use an open circle on the number line Step 3: Draw a dark line on the number line in the direction of the numbers that make the inequality true. Place an arrow on the line you drew to signify that the line continues forever in that direction. Graphing Inequalities

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Examples: a)n > 4 b)h ≤ 16 c)k ≥ -3 d)n < -17 Graphing Inequalities You can also use the number line to determine if a value is a solution to an inequality. Simply look to see if the value given is part of the shaded region; if it is, then it is a solution. If it happens to be where the dot was drawn, then look to see if the dot is open or closed. An closed circle means that value is a solution… an open circle means it is not.

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Graphing Inequalities Is 9 a solution to the inequality represented by the graph? YES NO What is the inequality this graph represents? n ≥ -7 If you were to continue the number line it would extend to the 9 and therefore be a solution to the inequality

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Graphing Inequalities Is -6 a solution to the inequality represented by the graph? YES NO What is the inequality this graph represents? n ≥ -3 The graph of this inequality will never pass through -6 on the number line The inequality represented on this graph is n ≥ -3 and -6 is not greater than or equal to -3

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Graphing Inequalities Is 5 a solution to the inequality represented by the graph? YES NO If the circle was closed, then yes 5 would be a solution. An open circle means that it is simply “less than” in this case, not “less than or EQUAL to 5” What is the inequality this graph represents? n < 5

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COMPOUND INEQUALITIESEXTENSION Graphing Compound Inequalities Graphing compound inequalities is very similar to the procedures just discussed. However, when drawing the solutions line you must take into account what and “AND” statement and an “OR” state means. With “AND” statements the solution will be drawn in-between the two points plotted on the number line. Example: 4 < b ≤ 9

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COMPOUND INEQUALITIESEXTENSION Graphing Compound Inequalities Graphing compound inequalities is very similar to the procedures just discussed. However, when drawing the solutions line you must take into account what and “AND” statement and an “OR” state means. With “OR” statements the solution will be drawn in opposite directions. Example: b ≤ 4 or b > 9

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COMPOUND INEQUALITIESEXTENSION Graphing Compound Inequalities Practice: 1) 6 ≤ n < 9 2) 16 19 3) -3 < m < 5 4) 0 ≥ w or w ≤ -1

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Recently, you learned how to solve equations such as… 3x + 4 = 10 5 + 2n = 75 5v – 18 = 7 To solve an inequality, follow the same exact process as you did for the equations above. This time instead of having the equals (=), leave the inequality symbol (, ≤, or ≥). When you have finished solving the inequality, graph your solution. To check, pick a number that is on the line you drew on the graph and substitute that value into the original inequality. Make certain that you now have a true statement! Solving Inequalities?

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Solve each inequality and graph your solution. Then use your graph to check if your solution is accurate. a)9 + x > 81 b)3p – 25 < 14 c)3.036c + 6.2 ≥ 16.826 d). Solving Inequalities?

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COMPOUND INEQUALITIESEXTENSION Solving Compound Inequalities Once again, this will be very similar to the way we solved previous equations such as 3x + 6 = 15 or 8x – 9 = 7 If you are working with an “AND” statement, whatever you do one part of the inequality you must also do to all of the other parts of the inequality. Solve and graph the solution: 4 < x + 2 < 16

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COMPOUND INEQUALITIESEXTENSION Solving Compound Inequalities Once again, this will be very similar to the way we solved previous equations such as 3x + 6 = 15 or 8x – 9 = 7 If you are working with an “OR” statement, solve each portion of the inequality separately and then graph there solutions on the same number line Solve and graph the solution: 16 ≤ 4x or 18 > 3x

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COMPOUND INEQUALITIESEXTENSION Solving Compound Inequalities Practice: Solve and graph the solution. a) 10 < 9x + 1 < 19 b) 3b> 15 or 3b < 3 c) d) 3f + 1 > 25 or 6f – 8 ≤ 34

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