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CCSS.Math: ,

we are told we want to solve the following equation that the negative natural log of 2x is equal to 2 times the absolute value of X minus 4 all of that minus 7 one of the solutions is X is equal to 0.5 find the other solution they say hint use a graphing calculator and round your answer to the nearest tenth so pause this video and have a go of this if you like and then we'll work on this together and I encourage you to have a go on it a go go at it all right now let's work on this together now the key here is to realize that we might be able to solve this by graphing or at least approximate the solutions to this by graphing and the way we do that is if we have an equation especially a hairy equation like this in one variable we can set y equal to the left and then set y equal to the right and then graph each of those functions and then think about where they intersect because they'll intersect at an x value that gives us the same y value and that means that the two sides are the same so what do I mean by that well we could set Y is equal to the negative natural log of 2x so we could have one equation or one function like that and then we could have another equation or function that y is equal to 2 times the absolute value of x minus 4 minus 7 and let's see where they intersect and the X values where they intersect those are going to be solutions to that so I'm going to use desmos as my graphing calculator so let's type in the two sides so first I'll do the left side so if Y is equal to the negative natural log of 2x and actually let me make my color to be the same or as close as I can so maybe closer to that blue or color okay and then the next one I want Y is equal to 2 times 2 times the absolute value which I don't know whether desmos prefers I'll use that actually that works ok X minus 4 and then I will close my absolute value and then I have minus 7 and I will do this in the red color so that we can keep track of things okay so those are my two graphs and now I just need to think about where they intersect and one of the solutions is X equal to 0.5 that's not the one they want they want the other solution so to speak so let's see let's see we have one solution actually let me zoom in a little bit so when X is equal to 0.5 that's this solution that's this solution right over here it looks like Y is equal to 0 there but then the other point of intersection seems to be right over here and actually desmos has a nice little feature we're all elect it'll tell us that point right over there but you could even approximate it you can see that X is over 6 and then each of these let's see one two three four five each of those is 0.2 so it's going to be six point two something is what I would do and they want us to round to the nearest tenth anyway so you don't even need to use that feature but you can see very clearly that when X is equal to approximately six point two three eight that boat that we get a y is equal to negative two point five four or another way to think about it is when X is approximately equal to when X is approximately equal to six point two that the two sides of this equation are going to be equal to each other or approximately equal to each other and we're done we've just solved using a or at least approximated a solution using graphing and a graphing calculator