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CCSS.Math:

luis receives a gift card worth $25 to an online retailer that sells digital music and games each song costs $0.99 or 89 cents and each game costs a dollar 99 he wants to buy at least he wants to buy at least 15 items with this card set up a system of inequalities that represents this scenario and identify the range of possible purchases using a graph and that's why we have some graph paper over here so let's define some variables let's let s equal the number of songs he buys number of songs and then let's let G equal the number of games that he buys now if we look at this constraint right here he wants to buy at least 15 items with this card so the total number of items are going to be the number of songs plus the number of games and that has to be at least 15 so it has to be greater than or equal to 15 so that's what that constraint tells us right there and then the other constraint is the gift card is worth $25 so the amount that he spends on songs plus the amount that he spends on games has to be less than or equal to 25 so the amount that he spends on songs are going to be the number of songs he buys the number of songs he buys times the cost per song times 89 cents times I'll say zero point eight nine times s that's how much he spends on songs plus the cost per game which is a dollar ninety nine times the number of games this is going to be the total amount that he spends and that has to be less than or equal to twenty five less than or equal to 25 now if we want to graph these we first kind of have to define the axes so let me do that right here and we only care about the first quadrant because we only care about positive values for the number of songs in the number of games we don't talk about scenarios where he buys a negative number of songs or games so we're just going to be just the positive quadrant right here let me draw the axes so let's make this let's make the vertical axis that I'm drawing right here let's make that the vertical axis and and let's call that the song axis so that's the number of songs he buys let me make sure you can see that that is the song axis and then let's make this this horizontal that's going to be the number of games he buys just bowled it in the number number of games he buys number of games and just to make sure just to make sure that we can fit on this page because I'm a feeling we're going to get to reasonably large numbers let's make each of these boxes equal to 2 so this would be 4 8 12 16 20 so on and so forth and this would be 4 this obviously would be 0 4 8 4 8 12 16 20 and so on so let's see if we can graph these two constraints well this first constraint s + G is going to be greater than or equal to 15 the easiest way to think about this or the easiest way to graph this is to really think about the intercepts if G is 0 if G is 0 what is s well s plus 0 has to be greater than or equal to 15 so if G is 0 s is going to be greater than or equal to 15 let me put it this way so if I'm going to graph this one right here if G is 0 s is greater than or equal to 15 so G is 0 s 15 is reciept this is 12 14 15 is right over there and s is going to be all of the values equivalent to that or greater than for G equal to 0 if s is equal to 0 G is greater than or equal to 15 so if s is equal to 0 G is greater than or equal to 15 so G is greater than or equal to 15 so the boundary line s + G is equal to 15 you would just have to connect these two dots let me try my best to connect these dots so it would look something like this this is always the hardest part see how well I can connect these two dots nope can you see actually get a line tool for this so that's pretty good so that's the line s plus G is equal to 50 and we talked about the values greater than 15 we're going to go above the line and you saw that when G is equal to 0 s is greater than or equal to 15 it's all of these values up here and when G when s was 0 G was greater than or equal to 15 so this constraint right here is all of this all of this area satisfies this all of this area if you pick any coordinate here it represents and really you should think about the integer coordinates because we're not going to buy parts of games but if you think about all of the integer coordinates here they represent combinations of s and G where you're buying at least 15 games for example here you're buying 8 games and 16 songs that's 24 so you're definitely meeting the first constraint now the second constraint 0.89 s plus 199 G is less than or equal to 25 this is a starting point let's just draw the line let's draw the line 0.89 s plus 199 is equal to 25 and then we could think about what the less than what what region the less than would represent 199 G and the easiest way to do this once again we could do slope y-intercept of all that type of thing but the easiest way to just find the s and the G intercepts so if s is equal to 0 if s is equal to 0 then we have 199 G is equal to 25 so s is equal to 0 then we have 199 G is equal to 25 or G is equal to its going to calculator out for this so if we take 25 divided by 199 it is twelve point five six G is equal to twelve point five six let me get it twelve point five six and then if so when s is 0 let me plot this when s is 0 G is twelve point five six this is 12 this is 14 twelve point five six is going to be right right there a little bit more than 12 that's that value there and then let's do the same thing if G is 0 so if G is equal to 0 then we have so this term goes away we have zero point 8 9 s if we use just the equality here the equation is equal to 25 or s is equal to calculator out again so if we take 25 divided by 0.8 9 we get it's equal to 28 point zero eight just a little over twenty eight so twenty-eight point zero eight so that is G is 0 s is 28 so that is two for twenty four six eight little over 28 so it's right over there so this line 0.89 s plus 199 G is equal to 25 is going to be it's going to go from this coordinate which is 0 28 so that point right there all the way down to the point twelve point five six zero so let me see if I can draw that I'm going to go and draw one more attempt maybe if I start from the bottom it'll be easier that was a better attempt let me Bowl that in a little bit so you can make sure you can see it so that line represents this right over here now if we're talking about the the less than area what will that imply so when if we think about it when G is equal to zero point eight nine s is less than 25 so when G is equal to zero if we really wanted the less than there we could think of it this way it's less than instead of just doing less than or equal to so s is less than 28 point oh eight so it'll be the region below it'll be the region below when s is 0 G so if we think s is zero if we use this original equation $1.99 G will be less than or equal to I use this just to plot the graph but if we actually care about the actual inequality we get $1.99 G is less than 25 G would be less than or equal to twelve point five six so when s is equal to G zero G is less than twelve point five six so the area that satisfies this second constraint is everything below this graph everything below this graph now we want the region that satisfies both constraints so it's going to be the overlap of the regions that satisfy one of the two so the overlap is going to be this region right here below the orange graph and above the blue graph including including both of them so if you pick any combination so if he buys four if he buys four four games and 14 songs that would work or if we bought two games and 16 songs that would work so you can kind of get the idea anything in that region and you can only buy integer values would satisfy both constraints