In Exercises 15–35, solve each equation. Then state whether the e...

Question

In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4

Accepted Answer

Hello. Today we're gonna go ahead and solve a rational equation and figure out whether that equation is identity, conditional or inconsistent. So the rational equation given to us is five x plus 4/ -14/3 Equal to X -1/6. Now since we want to sell for X let's go ahead and set the equation equal to zero. In order to do that. All I have to do is just take this quantity and subtract it over to the other side. Doing so is going to give me five X plus 4/9 minus 14/3 minus x minus 1/6. Equal to zero. Now I want to go ahead and combine all these quantities together in order to do that I need to figure out the lowest common denominator between 93 and six. The lowest common denominator for these three numbers is going to be 18 in order to get 18 on all three of them I need to multiply the first one by two, the second one by six and the third one by three doing so is going to give me two times five X plus 4/18 minus six times 14/ minus three times x minus 1/18 And all of that is going to equal to zero. Now all of these are under a common denominator that means that we can go ahead and combine them all into a single fraction. Doing so is going to give me two times five X plus four minus six times 14 minus three times x minus one. All of that is going to be under 18 and all of that is going to equal to zero. Now I can go ahead and get rid of the denominator here, if I go ahead and multiply everything by 18. Doing so will leave me with two times five X plus four minus six times 14 minus three times X minus one equal to zero. And now what we need to do is just solve for X In order to do that. Let's go ahead and open up the parentheses in order to do that. I'm gonna distribute to to every term here, multiply six and 14 together and distribute negative three to every term here. Doing so is going to give me 10 x plus eight minus 84 - x plus three equal to zero. And now I'm gonna go ahead and just combine like terms together, I can go ahead and combine 10 X and negative three X. Doing so is going to give me positive seven X. And I can go ahead and combine eight, positive eight, negative 84 positive three. Doing so is going to give me negative and that's going to be equal to zero. Now in order to solve for X. All I need to do is just add 73 to both sides. Doing so is going to give me seven X equal to 73. And in order to solve for X I just need to multiply divide, I'm sorry, divide both sides by seven And that's going to give me X is equal to 73/7. So we only have one solution for X. Recall that if you only have one solution for an equation, your equation is conditional, that means that the solution to this problem is going to be a. So I hope this video helps you reduce rational equations and figure out whether an equation is conditional or something else. And I'll see you all in the next video.

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