Exercises 41–60 contain rational equations with variables in deno...

Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation.

3/(2x - 2) + 1/2 = 2/(x - 1)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this rational equation we are to solve for it and indicate the value. That makes the denominator zero. Alright, so we've got nine divided by three H minus six plus 9/ equals 12 over H minus two. Alright, now, before we get any further let's take a look at that denominator and see if we can do any factoring well with this 1st 13 H minus six, we could factor out of three. So that would be three times H minus two. Cool. And now we want to pause for a minute and think about what would make our denominators equal to zero? What would make them undefined? And therefore just something we can't have. We cannot abide by. Well let's set these factors these H minus twos equal to zero to see what number we would have to put in for H to make it zero, we'll add two to both sides. So if we had a church equals two and put that in like this we'd have a zero in the denominator can't have that. Therefore h cannot equal to. All right now let's go ahead and solve for H looking at this, we want to say Let's get rid of the fraction. And what would our common denominator be? What is something that all of those denominators go into? Well the H -2 factor for sure. And then what else? We've got to include this three and this two. So three times two. Well that's six. So let's multiply everything by six times H minus two. So the nine from our first one now times six and h minus two all over. And I'll write this is the factored form three times H minus two plus this second nine times six times H minus two. All over. Just A two equals times our six times H minus two. And that is all over H minus two. Now the fun part let's see what cancels out what factors are in both the numerator and the denominator. Well in the first one we've got a six and a. Three and that six divided by three is going to leave two in the numerator and the H minus two's those cancel out now for this second one we've got that six divided by two and six divided by two. That leaves us with three. And over here on the right hand side of the equation this h minus two cancels out with that h minus two. So now what have we got on the left hand side, we've got nine times two plus nine times three times H minus two. On the right hand side. We've got 12 times six. Alright let's do what multiplication we can nine times two is 18 plus nine times three is 27. And that will still need to get distributed to that h minus two and that equals 12 times six. Well that's 72. Now let's go ahead and do that distribution. We've got 18 plus 27 times ages H 27 times negative two is negative 54 equals 72. Now let's combine our like terms on the left hand side we've got 18 and a minus 54. So 18 minus 54 gives us negative 36 plus 27 H. Equals 72. Let's work on getting that H. By itself. We will add 36 to both sides. And on the left we've got 27 h. And on the right now we've got We can get that H. by itself by dividing both sides by 27. And on the left hand side that's just A. One H. Or just H. Equals. And on the right hand side 108 divided by 27 gives us four. So we look really quick and make sure that this is not the same as what H cannot be great. H. Equals for is not two. So we know we can proceed we look at our answer choices and that matches with answer choice. D. Well done we'll catch you on the next one.

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 3/(2x - 2) + 1/2 = 2/(x - 1)

Key Concepts

Rational Equations

Restrictions on Variables

Solving for Variables

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