Inequalities

1. Inequalities

2. How Do You Read an Inequality? > 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 “nine is greater than three times a number” “four times a number is less than nineteen”

3. 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”

4. How Do You Read an Inequality? • Examples: • 14 > 2n ______________________________________ • 15 ≥ 3 + n ____________________________________ • 23 ≤ 30 – n ___________________________________ • 12 < n ÷ 9 ___________________________________ • 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 twelve is less than a number divided by nine nine plus a number is less than twenty-four minus that same number.

5. Is It a Solution? 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? 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.

6. Is It a Solution? Which of the following inequalities is/are incorrect when n = 10? 6n + 8 > 68 5(6 + n) ≥ 80 – 10 < 0 ≤ 9 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

7. Which Symbol Should Be Used? Is more than Is greater than Is larger than Is above Is bigger than Is less than Is smaller than Is below Is at least Is no less than Is no smaller than Is at minimum Is at most Is no more than Is no greater than Is at maximum

8. Translating Inequalities 5 > n five is greater than a number: _____ sixteen is less than or equal to a number: ______________ 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: ______________ 16 ≤ n 2n – 4 ≥ 75 n – 14 ≥ 17

9. Is It a Solution? 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. 16n ≥ 120 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. n

10. Is It a Solution? Keith has $500 in his savings account at the beginning of June. He wants to have more than $200 in the account on December 1st so that he can purchase holiday gifts for his family and friends.Keith withdraws $75 each month for his own expenses. Write an inequality to represent Keith’s situation. December is 6 months away… will he meet his goal? Explain. 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.

11. Is It a Solution? 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. Write an inequality to represent Kelly’s situation. If Kelly collects an additional 29.1 pounds of food will she need an additional box? 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.

12. Is It a Solution? 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. Write an inequality that represents Martina’s situation. If Martina earns a 96% on her next test will she meet her goal? Explain. 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.

13. Graphing Inequalities • 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 • For < and > 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.

14. Graphing Inequalities Examples: n > 4 h ≤ 16 k ≥ -3 n < -17 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.

15. Graphing Inequalities Is 9 a solution to the inequality represented by the graph? YES NO What is the inequality this graph represents? n ≥ -7

16. Graphing Inequalities Is -6 a solution to the inequality represented by the graph? YES NO What is the inequality this graph represents? n ≥ -3

17. 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

18. Solving Inequalities? 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!

19. Solving Inequalities? Solve each inequality and graph your solution. Then use your graph to check if your solution is accurate. 9 + x > 81 3p – 25 < 14 2.6565c + 6.2 ≥ 16.826 .