EnVision Algebra 2 Lesson 4 5 Solving Rational Equations Critique Explain Examples 1 2

hi welcome to Algebra 2 lesson 4-5 solving rational equations in this lesson we'll be able to solve rational equations and identify extraneous solutions what are extraneous solutions that's part of the vocabulary rational equation extraneous solution are the vocabulary that we will stress in this lesson let's start with critique and explain Nikki and Tavon use different methods to solve the equation 1 / 2X + 2 over 5 = 9 over 10 okay explain the different strategies that Nikki and ton used and the advantages or disadvantages of each so let's look at Nikki's strategy here we have 1 2x + 2 over 5 = 9/ 10 we start with the same thing Nikki did subtraction first to add the like terms of the con and and simplified by solving for Tavon however what did he do he multiplied 10 which is the LCM of the denominator and and so he got rid of the fractions and then he solved the equation without fraction and the solutions are the same the solution is one so different methods but still the same solution right so this is basically what you need to explain the different strategies they used we can we can summarize and say that Nikki subtracted 2 over 5 to combine the constants and solved for Tavon however multiplied the LCN of the denominators to simplify the whole equation then solved for so Part did Nikki use the correct method to solve the equation didon yes both used correct methods their Solutions are the same why might Tavon have chosen to multiply both sides of the equation by why would we do this what is the benefit of doing this well we could get rid of the fractions right by doing so he gets rid of the fractions could he have used another number yeah sure but this is the LCM which would be the most simple number so that our numbers are as simple as possible but of course we could find another another multiple of two five and 10 which is 20 he could use 20 but the numbers or the but the numbers and coefficients are bigger they get bigger okay so in this lesson we're going to ask this essential question how can you solve rational equations and identify extraneous solutions let's look at example one solve rational equation what is the solution to each rational equation rational equation is an equation that that contains rational expression so if you see the equal sign like this it is an equation and if you see rational expression like this one in the equation it is rational equation so let's look at this rational equation in part 1 /x + 4 is = 2 in order to solve this you want to first figure out how to bring your variable into the numerator because in order to solve for your variable you need to have in the numerator you you want over one equals to to certain answer right so in order to do that we're going to multiply both sides of the equation by the same denominator just like how we would do to solve just equations regular equations without without rational equation but still like with with rational numbers right so multiply the denominator which is the expression + 4 on both sides so that we balance our equation our equation should still remain the same so that we're not changing the equation to get solution for different equation right we want solution for this equation so then you can cancel this out so 1 is equal to 2x + 8 2 * 4 which is equal to 2x is equal to 1 - 8 and so is equal to -7 /2 so the solution is -7 /2 okay very easy and then Part you do the same thing can multiply denominator the expression in the denominator and then cancel out would you can simplify simplify solve for and that's your solution so let's look at try number one and what's the solution for each equation in and solve the rational equations by yourself pause the video come back when you're ready for answers okay are you ready let's look at part 2 over + 5 is equal to 4 so you're going to multiply + 5 on both sides and then you can cancel this out so it's equal to 2 * 4x + 20 then if you solve for 2 - 20 is8 so is equal to8 over 4 simplify that's going to be -9 /2 so 9 over two and then Part do the same thing multiply the denominator on both sides cancel this out 1 is = 2x - 14 so 2x is = 1 + 14 15 is = 15 / 2 check your answer see if you got it right if you did good job okay let's look at example two solve work rate problem Arthur and cheyan can paint wall in 6 hours when working together cheyan works twice as fast as Arthur how long would it take shyenne to paint the wall if she were working alone so Cheyenne and Arthur together work for six hours right and we know she works twice as fast as Arthur so this this is double she gets double work done in the in the same hour right so first of all we're going to figure out Expressions how to write expressions to represent the situation so we're going to let variable represent the number of hours Arthur needs to paint the wall himself so he needs amount of time so cheyen Works double as much so that means she is going to work twice as much so she's faster so she needs half of the hour that he works to finish one work so in other words Arthur can paint one wall in hours or it's 1 /x of the whole wall in one hour does that make sense so he can paint one wall this is wall in XI hours after hours he paints everything but if we divide this by the amount of hours he worked that means this is 1 /x of the entire wall right so 1 /x of wall in 1 hour is the rate that he works so Cheyenne is twice as fast so she paints two over of wall in one hour so that's rate and then together using the distance equation is equal to RT well in this case is the is the wall okay so the wall represents the whole wall represents one so here you can just represent this as one so one is equal to together the rate is this plus this right right so because the rate is combined right so the rate whoops this is not the rate 1 /x is Arthur's plus 2 overx is shean and then they together works six hours to paint the entire thing right so here in this equation using this equation we can we can figure out so dividing six on both sides we get this equation in step two 1x + 2x is = to 1 over six and then we can use yeah we can we can multiply we can multiply on both sides or we can add 1 + 2 right 3 overx 3 overx is equal to 1/ 6 and then we can multiply on both sides and then we can multiply six on both siid to get is = to 3 * 6 which is 18 so together Arthur is working 18 hours and then Cheyenne is working yeah 18 / 2 that's the rate yeah so it Tak takes Arthur 18 hours to paint the wall alone if he were to do it by himself this is the rate that he's this is the time it's going to take for him to paint the whole wall by himself right and then for for chenne she will probably paint it in 9 hours because she's twice as fast yeah so using the first Expressions you can plug in to Fig fig it out is 18 so Arthur is going to need 18 hours is 18 so 18 / by 2 is 9 so cheyen is going to need 9 hours so use the distance equation but it works for the project as well would would also equal to one project or one thing that you need to be done or fail let's look at turn number two it takes 12 hours to fill pool with two pipes where the water in one pipe flows three times as fast as the other pipe how long will it take the slower pipe to fill the pool by by itself so pause the video see if you can do this by yourself come back when you're ready for answers okay are you ready so it's very similar to example two it takes two hours to fill pool with two pipes the two pipes have different rates right so use the distance equation where is really filling the pool and so this represents the whole pool which is going to be represented by one so filling the whole pole is one okay and then rate if if if filling the whole thing is one if distance is one right then the rate is going to be one divided by how many hours it took for each pipe right so water of one pipe flows three times as fast so rate of the faster one is going to be one the whole the whole pool divided by three times as fast as the other type so if we say the slower one rate of the slower one is 1 divided by right is the time that it takes for for it to fail the whole thing the the faster one takes three times as fast so 1 /x * 3 so that's going to be 3 overx and these are the different rates for each one so using the distance equation again this is the project of the pool this is the pro this is the pool represented by one and then the rate together 3x + 1 /x is and times the time that it takes so 12 hours right is going to be complete right so solve for using this equation it means we have 1 over 12 is equal to 3 + 1 which is 4X and so multiplying on both sides and 12 on both sides we get is equal to 4 * 12 which is equal to 48 and this is terms of hours so the slower pipe this is the rate so the time is just okay so it's going to take 48 hours to for the slower pipe to fill the pool by itself okay so let's continue with the next examples in the next video