Solving absolute value inequalities: fractions

we have the absolute value of 2 R minus 3 and 1/4 is less than 2 and 1/2 and we want to solve for R so right from the get-go we have to deal with this absolute value and just as a bit of a review if I were to say that the absolute value of x is less than well let's just say less than 2 and 1/2 that means that the distance from X to 0 is less than 2 and 1/2 that means that X would have to be X would have to be less than 2 and 1/2 and X would have to be greater than negative 2 and 1/2 and just think about it for a second if I were to draw it on a number line right here if I were to draw it on a number line that is 0 that is 2 and 1/2 and that is negative 2 and 1/2 these two numbers are exactly 2 and a half away from 0 because both of their absolute values is 2 and a half now if we want all of the numbers whose absolute value is less than 2 and 1/2 or that are less than 2 and 1/2 away from 0 it would be all of the numbers in between it would be all the numbers in between and that's exactly what these two statements are saying X has to be less than 2 and 1/2 and it has to be greater than negative 2 and 1/2 if this absolute value were the other way the absolute value of x has to be greater than 2 and 1/2 that would be the numbers outside of this and it would be an or but we're dealing with a less than situation right there so let's just do what we were able to figure out when it was just an X the distance from this thing the distance from this thing to 0 has to be less than 2 and 1/2 so we can write we can write that 2r minus 3 and 1/4 can either well it has to be less than 2 and 1/2 and and 2r minus 3 and 1/4 has to be greater than negative 2 and 1/2 same exact reasoning here let me draw a line so we don't get confused same exact reasoning here this quantity right here has to be between negative 2 and 1/2 it has to be greater than negative 2 and 1/2 right there greater than negative 2 and 1/2 and it has be less than two and a half so that's all I wrote there so let's solve each of these independently well this first one over here you've learned before that I don't like well I don't like improper fractions and I don't like fractions in general so let's make all of these fractions or sorry I don't like mixed numbers I want them to be improper fraction so let's turn all of these into improper fractions so if I were to rewrite it we get 2 R minus 3 and 1/4 is the same thing as 3 times 4 is 12 plus 1 is 13 - R - thirteen - thirteen fourths is less than two times two is four plus one is five is less than five halves so that's the first equation and then the second equation and do the same thing here we have 2 R minus 13 over four has to be greater than negative five-halves all right now let's solve each of these independently to get rid of the fractions the easiest thing to do is to multiply both sides of this equation by 4 that'll eliminate all of the fractions so let's do that let's multiply let me scroll to the left a little bit let's multiply both sides of this equation by 4 and what do we get 4 times 2 R is 8r 4 times negative 13 over 4 is negative 13 is less than and I multiplied by a positive number so I didn't have to worry about swapping the inequality is less than 5 halves times 4 is 10 right it becomes you get a 2 and a 1 it's 10 so you get 8 R minus 13 is less than 10 now we can add 13 to both sides of this equation so that we get rid of it on the left hand side a dirty non both sides and we get 8r these guys cancel out is less than is less than 23 and then we divide both sides by 8 divide both sides by 8 and once again we didn't have to worry about the inequality because we're dividing by a positive number and we get R is less than 23 23 over 8 or if you want to write that as a mixed number R is less than was that two and seven two and seven eighths so that's one condition but we still have to worry about this other condition there was an and right here let's worry about it so our other condition tells us to our minus 13 over 4 has to be greater than negative five-halves let's multiply both side of this equation by 4 so 4 times 2r is 8r 4 times negative 13 over 4 is negative 13 is greater than negative five-halves times 4 is negative 10 now we add 13 to both sides of this equation add 13 to both sides of this equation the left-hand side these guys cancel out you're just left with 8 R is greater than negative 10 plus 13 is 3 or divide both sides of this by 8 and you're left with R has to be greater than 3/8 so our two conditions all has to be less than 2 and 7/8 and greater than 3/8 or we could just write it like this R is greater than 3/8 so it's greater than or maybe I should say 3/8 is less than R which is less than 2 and 7/8 2 and 7/8 so if we were to plot the solution on the number line so I'm about to do so that's my number line this is zero right here maybe this is 1 2 & 3 we have 2 and 7/8 so we have to be less than 2 and 7/8 let's say that this is 2 and 7/8 right there and we have to be greater than 3/8 say that is 3/8 so 3/8 will be someplace right around there and everything in between is a valid solution everything in between is a valid solution and we could try it out let's try out something that based on what I just drew should be a valid solution 1 should be a valid solution let's try it out here 2 times 1 let's do it 2 times 1 minus 3 and 1/4 what is that that's 2 minus 3 and 1/4 and so what is that 2 minus 3 and 1/4 is well 3 and 1/4 minus 2 is 1 and 1/4 so this will be negative 1 and 1/4 negative 1 and 1/4 we're taking the absolute value of it so we take the absolute value of it which is equal to 1 and 1/4 which is indeed less than 2 and 1/2 now let's try another number let's try 0 based on this zero should not work so what happens if we put 0 here you get 2 times 0 which is 0 minus 3 and 1/4 you take the absolute value of minus of negative 3 and 1/4 look at positive 3 and 1/4 which won't work 3 and 1/4 is greater than 2 and 1/2 so that work that's true that works out and the same thing for 3 2 times 3 is 6 minus 3 and 1/4 is 2 and 3/4 take the absolute value it's 2 and 3/4 it's still bigger than 2 and 1/2 so it won't work so at least the points that we try it out seem to validate this solution that we got