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/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this big rational equation and we are to solve it and indicate the values that make the denominator zero. So we've got two divided by X plus one plus one half divided by x minus one equals one over X plus one times x minus one. So now let's pause and recall from previous lessons when we want to find out what will make the denominator zero, we're going to set those denominators equal to zero. So it looks like our only factors here are X plus one And X -1. So we set those both equal to zero to find out what numbers we could put in for X to make it zero. So on this first one we'll subtract one from both sides. So we get an X equals negative one and then this one will add one to both sides and we get an X equals one. So what does this tell us? Well if we put a negative one or one in for X then we're going to get a zero in the denominator which is undefined, we can't have that. Therefore X cannot equal negative one or one. Those are the values that make the denominator zero. So now let's solve for X. We can look at this and we can say what is our common denominator to get rid of this fraction because yuck this fraction and our common denominator is going to be these two factors X plus one times X minus one. And now we can rewrite this, we've got two times what that's being multiplied by is the X plus one X minus one in the numerator and that's over X plus one plus one half being multiplied by this common denominator, X plus one X minus one. And that is over x minus one equals one times our X plus one X minus one. And that is over X plus one X minus one. Great. Now the fun part we get to look and see what factors cancel out. So in that first term we've got an X plus one in both the numerator and the denominator by by in the second one we've got an x minus one in the numerator and the denominator by by and over on the right hand side of the equation X plus one X plus one X minus one. X minus one in both the numerator and the denominator. Good bye. So we've got two times x minus one plus one half times X plus one equals one. Now let's distribute to get rid of these parentheses. So instead of two times X minus one we've got two x minus two times one is two plus one half times X is one half X plus one half times one is one half equals one. And now we can let's see we've got a fraction. We could go ahead and again say what's our common denominator and multiply everything times that common denominator is two. So now we've got a two X times two -2 times two plus one half times two. Oops. Close my parentheses there one half X times two plus a one half times two. Oh no I did it again. Close that parentheses goodness gracious And that equals a one times two. And now we'll do that. Multiplication two X times two is four X minus two times two is four plus two times one half X is going to be X plus two times one half is going to be one equals two times one which is two. Now let's combine like terms, we've got a four X plus and X. And we've also got a negative four plus a one four X plus X gives us five X negative four plus one is minus three equals two. Now let's bring that negative three over to the right hand side so we'll add three to both sides. And on the left hand side we've got five X. And on the right hand side we've got five. Now let's get that X by itself will divide both sides by five. On the left hand side five divided by five is one. So that's just X. And on the right hand side, five divided by five. Well that's also one. So X equals one. Well that's really exciting except hang on just a minute here. What did we say at the beginning X cannot equal negative one or one. And we just said X equals one. If we stuck in a one in there for X, we would have it be undefined. Look at this, that would be a one minus 1/0 for that fraction. Oh, no. So we look at our answer choices and that is answer choice. D. No solution. Well done, We'll catch you on the next one.

Key Concepts

Rational Equations

Restrictions on Variables

Solving Rational Equations

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