Solving Inequalities


 Albert Sherman
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1 Solvig Iequalities Say Thaks to the Authors Click (No sig i required)
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3 Cocept 1. Solvig Iequalities CONCEPT 1 Solvig Iequalities Itroductio The Movie Tickets Marc ad Kara have made several frieds swimmig at the tow pool. Oe raiy day, they decided to ivite the group to go to a movie. Everyoe is very excited. Gradma is so pleased that Kara ad Marc have bee makig frieds that she has offered to treat. How may of you are goig?she asks Kara whe she hears of the pla. Well, there are eight of us altogether, Kara said. Alright, let me see, Gradma says reachig ito her wallet. I have fortyeight dollars to cotribute. Here it is. Thak you Gradma, the twis add smilig. Later at the movie theater, Kara takes out the moey. They have $48.00 to sped o tickets. Last miute, oe of the frieds has brought her brother alog. Kara ist sure that they have eough moey for the brother too. If each ticket is $6.00, how may tickets ca she buy without goig over the $48.00? This problem requires a iequality. The total must be less tha or equal to $48.00 without goig over. The movie is goig to start soo. Pay attetio to the work i this lesso ad you will kow how to help Kara purchase the tickets at the ed of it. What You Will Lear By the ed of this lesso, you will have leared how to complete followig skills. Graph iequalities o a umber lie. Recogize equivalet iequalities. Solve iequalities ad graph solutios. Model ad solve realworld problems ivolvig iequalities. Teachig Time I. Graph Iequalities o a Number Lie I the last few lessos you have bee learig about equatios ad about balacig a equatio. Lets thik about equatios. A equatio is a umber statemet with a equal sig. The equal sig tells us that the quatity o oe 1
4 side of the equatio is equal the quatity o the other side of the equatio. We ca solve a equatio by figurig out the quatity that will make the equatio a true statemet. x + 5 = 12 If we thik about this equatio, we ca use metal math ad we kow that the ukow quatity is equal to 7. If we substitute 7 i for x, we will have a true statemet = = 12 Our equatio is balaced because oe side is equal to the other side. Notice that there is oe aswer for xthat makes this a true statemet. What is a iequality? A iequality is a mathematical statemet that ca be equal, but ca also be uequal. We use the followig symbols to show that we are workig with a iequality. >meas greater tha <meas less tha meas greater tha or equal to meas less tha or equal to How do we apply these symbols? Well, if you thik about it, we use the symbols to make a true statemet. Lets look at a example. x + 3 > 5 There are may possible aswers that will make this a true statemet. We eed the quatity o the left side of the iequality to create a sum that is greater tha five. Notice that the sig does ot have a lie uder it. We wat a quatity that is greater tha five ot greater tha or equal to five o the left side of the iequality. To make this true, we ca choose a set of umbers that has a 3 or greater i it. x = {3,4,5...} We dot eed to worry about solvig iequalities yet, the key thig to otice is that there are may possible aswers that will make a iequality a true statemet. We ca use graphs to help us uderstad iequalities i a visual way. Graphig iequalities o a umber lie ca help us uderstad which umbers are s solutios for the iequality ad which umbers are ot solutios. Here are some tips for graphig iequalities o a umber lie. Use a ope circle to show that a value is ot a solutio for the iequality. You will use ope circles to graph iequalities that iclude the symbols >or <. Use a closed circle to show that a value is a solutio for the iequality. You will use closed circles to graph iequalities that iclude the symbols or. 2
5 Cocept 1. Solvig Iequalities Write these hits i your otebook ad the cotiue with the lesso. Graph this iequality x > 3 To complete this task, first draw a umber lie from 5 to 5. The iequality x > 3 is read as x is greater tha 3. So, the solutios of this iequality iclude all umbers greater tha 3. It does ot iclude 3, so draw a ope circle at 3 to show that 3 is ot a solutio for this iequality. The draw a arrow showig all umbers greater tha 3. The arrow should face right because the greater umbers are to the right o a umber lie. The graph above shows the solutios for the iequality x > 3. The graph above ca help you see which umbers are solutios for x > 3 ad which are ot. For example, the arrow icludes the umbers 3.5, 4, ad 5. If you cotiued the umber lie, the arrow would also iclude the umbers 10 ad 100. So, all those umbers3.5, 4, 5, 10, ad 100are possible values for x. Graph this iequality x < 1. First, draw a umber lie from 5 to 5. The iequality x < 1 is read as x is less tha 1. So, the solutios of this iequality iclude all umbers less tha 1. It does ot iclude 1, so draw a ope circle at 1 to show that 1 is ot a solutio for this iequality. The draw a arrow showig all umbers less tha 1. The arrow should face left because the lesser umbers are to the left o a umber lie. The graph above shows the solutios for the iequality x < 1. Graph this iequality x 0. First, draw a umber lie from 5 to 5. The iequality x 0 is read as x is greater tha or equal to 0. So, the solutios of this iequality iclude zero ad all umbers that are greater tha 0. Draw a closed circle at 0 to show that 0 is a solutio for this iequality. The draw a arrow showig all umbers greater tha 0. 3
6 The graph above shows the solutios for the iequality x 0. 7M. Lesso Exercises Aswer the followig true/false questios. 1. True or false. A ope circle o a graph meas that the umber is ot icluded i the solutio set. 2. A iequality ca ever be equal. 3. A closed circle o a graph meas that the umber is icluded i the solutio set. Take a few miutes to check your aswers with a fried. II. Recogize Equivalet Iequalities Sometimes, you may eed to rewrite a iequality as a equivalet iequality i order to better uderstad it. This meas that a iequality ca be writte i two differet ways so that we ca uderstad it. Look at the example below. Graph this iequality 4 x. Draw a umber lie from 5 to 5. The iequality 4 x is read as 4 is greater tha or equal to x. This iequality will be easier to uderstad if we rewrite it so that the variable is listed first. If we list the x first, we must reverse the iequality symbol. That meas chagig the greater tha or equal to symbol ( ) to a less tha or equal to symbol. 4 x is equivalet to x 4. This makes sese. If 4 is greater tha or equal to x, the x must be less tha or equal to 4. The iequality x 4 is read as x is less tha or equal to 4. So, the solutios of this iequality iclude 4 ad all umbers that are less tha 4. Draw a closed circle at 4 to show that 4 is a solutio for this iequality. The draw a arrow showig all umbers less tha 4. The graph above shows the solutios for the iequality x 4. 4
7 Cocept 1. Solvig Iequalities Write a equivalet iequality for x 5. To write a equivalet iequality, we reverse the terms. If x is greater tha or equal to five the five is less tha or equal to x. Lets rewrite this. The equivalet iequalities are x 5 ad 5 x. 7N. Lesso Exercises Write a equivalet iequality for each example y 2. x > 7 3. a < 4 Check your aswers with a parter. III. Solve Iequalities ad Graph Solutios Now that you have developed a uderstadig of iequalities we ca solve them ad graph the solutio sets. We ca solve a iequality i a similar way as we would use to solve a equatio. The tricky part is i the aswer ot i the process. Oce we solve the iequality, we ca graph the solutio. Lets look at a example. Solve this iequality ad graph its solutio 4 3. Solve the iequality as you would solve a equatio, by usig iverse operatios. Sice the 4 is subtracted from, add 4 to both sides of the iequality to solve it. You do ot eed to multiply or divide both sides by a egative umber, so you do ot eed to reverse the iequality symbol. The symbol should stay the same ( 4 + 4) Now, graph the solutio. The iequality 7 is read as is less tha or equal to 7. So, the solutios of this iequality iclude 7 ad all umbers that are less tha 7. Draw a umber lie from 0 to 10. Add a closed circle at 7 to show that 7 is a solutio for this iequality. The draw a arrow showig all umbers less tha
8 The solutio for this iequality is 7, ad its graph is show above. Solve this iequality ad graph its solutio 2 < 14. Solve this iequality as you would solve a equatio, by usig iverse operatios. Sice the is multiplied by the, divide both sides of the iequality by to solve it. Sice this ivolves multiplyig both sides of the iequality by a egative umber, the sese of the iequality will chage ad you will eed to reverse the iequality symbol. This meas chagig the iequality symbol from a less tha symbol (<) to a greater tha symbol (>). 2 < > > 7 > 7 Now, graph the solutio. The iequality > 7 is read as is greater tha or equal to 7. So, the solutios of this iequality iclude all umbers that are greater tha 7. Draw a umber lie from 10 to 0. Add a ope circle at 7 to show that 7 is ot a solutio for this iequality. The draw a arrow showig all umbers greater tha 7. The solutio for this iequality is > 7, ad its graph is show above. Sometimes, you will eed to take more tha oe step to solve a iequality. You ca thik of these problems i the same way that you thought about twostep equatios Solve this iequality as you would solve a equatio, by usig iverse operatios. First, try to get the term with the variable, 3, by itself o oe side of the iequality. Sice the 9 is beig added to 3, subtract 9 from both sides of the iequality. You do ot eed to multiply or divide both sides by a egative umber, so you do ot eed to reverse the iequality symbol durig this step ( 9 + 9) There is a secod step you must take to fid the solutio. Sice is divided by 3, you must multiply both sides of the iequality by 3 to fid its solutio. This ivolves multiplyig by a positive umber, 3, so you do ot eed to reverse the iequality symbol. Be careful! It is true that you will eed to multiply 3 by 18 to fid the solutio. 6
9 Cocept 1. Solvig Iequalities However, sice you are ot multiplyig both sides of the iequality by a egative umber, you do ot reverse the iequality symbol. The solutio for this iequality is 54. 7O. Lesso Exercises Solve each iequality. 1. x 4 < y x Check your aswers with a fried. The cotiue with the ext sectio. IV. Model ad Solve RealWorld Problems Ivolvig Iequalities Now that you kow how to graph ad solve iequalities, let s take a look at some ways we ca use iequalities to solve problems. We ca use iequalities to represet some realworld problem situatios, too. Let s take a look at some key words that ca help us write iequalities to represet realworld problems. TABLE 1.1: > < greater tha more tha less tha fewer tha greater tha or equal to at less tha or equal to at least most The key words above provide clues about which iequality symbol you should use to represet a problem situatio. While key words ca be a helpful guide, it is importat ot to rely o them totally. It is always most importat to thik about what traslatio of the problem makes the most sese. This is especially importat because the same key words may mea differet thigs. For example, the key words more tha may mea you should use a >symbol or they may mea you should write a additio expressio. 7
10 Atoio is buyig milk for a breakfast evet. Each cotaier of milk costs $3. At most, he ca sped $12 o milk for the evet. a. Write a iequality to represet, c, the umber of cotaiers of milk he ca buy. b. Could Atoio buy 4 cotaiers of milk for the evet? Explai. Cosider part a first. Use a umber, a operatio sig, a variable, or a iequality symbol to represet each part of the problem. Sice each cotaier of milk costs $3, you ca fid the total cost, i dollars, of the milk he buys by multiplyig 3 by the umber of cotaiers. The key words at most idicate that you should use a symbol. Each cotaier... costs $3. At most, he ca sped $12... c 3 12 You may also wat to cosider that the value of c must be a iteger greater tha or equal to 0. Thik about why that is for a momet. The reaso that the value of c must be a iteger greater tha or equal to zero is because Atoio caot buy a egative umber of cotaiers or ca he buy a fractio of a cotaier. Neither of those situatios makes sese i real life. Whe usig iequalities to represet reallife situatios, you should always thik about which values would make sese for the variable ad which values would ot make sese. Next, cosider part b. Solve the iequality to help you. 8
11 Cocept 1. Solvig Iequalities c 3 12 c c c 1 4 c 4 Accordig to the iequality above, the umber of cotaiers, c, that he ca buy must be less tha or equal to 4. Sice 4 is a solutio for this iequality, ad sice 4 is a iteger that is greater tha 0, he could buy 4 cotaiers of milk. At the Stereo Store, Erika bought a $9 set of headphoes ad a DVD that is o sale for half its regular price. Erika spet more tha $15 o these two purchases. a. Write a iequality to represet d, the regular price of a DVD at the store. b. List three possible values of d. Cosider part a first. Use a umber, a operatio sig, a variable, or a iequality symbol to represet each part of the problem. The key words more tha, i this case, idicate that you should use a >symbol. $9 set of headphoes ad a DV D...for half the regular price...spet more tha $ d 2 > 15 Next, cosider part b. Solve the iequality to help you. 9
12 9 + d 2 > d 2 > d 2 > 6 d 2 > 6 d 2 2 > 6 2 d > 12 d 1 > 12 d > 12 Accordig to the iequality above, the regular price of a DVD, d, is more tha 12 dollars. So, three possible values of d are $12.50, $13, ad $20. These are oly 3 possible aswers. You could choose ay amout that is greater tha 12 dollars. Now lets go back to our itroductio problem ad help Toy with his dilemma o the trai. Real Life Completed The Movie Tickets Here is the origial problem oce agai. Use what you have leared about iequalities ad about solvig iequalities to help you with the movie tickets. Reread the problem before you begi ad uderlie ay importat iformatio. Marc ad Kara have made several frieds swimmig at the tow pool. Oe raiy day, they decided to ivite the group to go to a movie. Everyoe is very excited. Gradma is so pleased that Kara ad Marc have bee makig frieds that she has offered to treat. How may of you are goig?she asks Kara whe she hears of the pla. Well, there are eight of us altogether, Kara said. 10
13 Cocept 1. Solvig Iequalities Alright, let me see, Gradma says reachig ito her wallet. I have fortyeight dollars to cotribute. Here it is. Thak you Gradma, the twis add smilig. Later at the movie theater, Kara takes out the moey. They have $48.00 to sped o tickets. Last miute, oe of the frieds has brought her brother alog. Kara ist sure that they have eough moey for the brother too. If each ticket is $6.00, how may tickets ca she buy without goig over the $48.00? To figure this out, we eed to write a iequality. The origial group cosisted of 8 kids without the extra brother. Each ticket is $6.00. We ca use this i our iequality. If we have moey left over, the we will kow whether or ot we ca pay for him too. We eed to be less tha or equal to $ Here is the iequality. 6x 48 Now we ca solve the iequality. The x represets the umber of tickets that we ca purchase if they are $6.00 each. 6x 48 6x 6 48 x 8 This shows that the kids ca purchase less tha or equal to 8 tickets with the $ They dot have eough to pay for the brother too. Kara tells the group this iformatio ad they all chip i with their ow moey. All of the kids are able to atted the movie thaks to the geerosity of the group. Vocabulary Here are the vocabulary words that are foud i this lesso. Equatio a umber setece with a equal sig where the quatity o oe side of the equals is the same as the quatity o the other side of the equals. Iequality a umber setece where oe side is ot ecessarily equal to the other side. There are several possible aswers that will make a iequality a true statemet. Equivalet Iequalities Two iequalities that are writte differetly, but still express the same umber relatioships. Techology Itegratio 1. This is a video that explais how you ca solve a iequality. 11
14 Time to Practice Directios: For problems 14, graph each iequality o the give umber lie. 1. x < 3 2. x > Directios: For problems 56, solve each iequality ad the graph its solutio o the give umber lie. 5. x + 3 > 9 6. ( 4) 2 Directios: Solve each iequality. 7. x + 4 < x
15 Cocept 1. Solvig Iequalities 9. b a y > x y < x < x 2 = x x y a 7 > b + 9 < 39 Directios: Solve each problem. 21. Emma bought a fruit smoothie at a juice shop for t dollars, icludig tax. Emma paid with a $10 bill. She received less tha $5 i chage. a. Write a iequality to represet t, the umber of dollars, icludig tax, that Emma paid for the fruit smoothie. b. List three possible values of t. 22. Kiet has 16 juice boxes for a family picic ad eeds to buy more. Juice boxes are sold i packages of 8. a. Write a iequality to represet p, the umber of packages of juice boxes Kiet eeds to buy i order to have at least 40 juice boxes total for the picic. b. If Kiet buys 4 packages of juice boxes, will that be eough? 13
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