The equations in Exercises 79–90 combine the types of equations w...

Question

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation.

4x/(x + 3) - 12/(x - 3) = (4x^2 + 36)/(x^2 - 9)

Accepted Answer

Welcome back. I'm so glad you're here. We have got this big equation here and we're to determine whether it's identity conditional or inconsistent. Well, what do each of those mean? Let's look at our answer choices. A conditional equation. We recall from previous lessons that a conditional equation. That's where you have X equaling at least one real number. And that's what we get most of the time when we're solving problems, X could equal one or X could equal a million, but X equals at least one real number with an identity equation. X actually equals all real numbers. And what that looks like it could be X equals X. You could have something like one equals one. Both sides of the equal side sign are going to be the same. They'll be identical an inconsistent equation while that's where X equals no real numbers. And it's going to look nonsensical. You could have something like X equals X plus one or you could have something like zero equals one. That's just not true. Zero doesn't equal one. So that is inconsistent. So let's see what we've got here. We've got nine X divided by X plus seven plus 63 divided by X minus seven equals X. R. Nine X squared plus divided by X squared minus 49. Well, we've got fractions. We want to clear those fractions and we ask ourselves what can we do to clear these fractions, what would our common denominator B While we look at the right hand side, that denominator we know that's a difference of squares and it could be factored that first square X squared? That's X times X. And that second square 49. That's seven times seven, one's positive, one's negative. And hey look, those are the same factors. The same denominators on the left hand side. So our common denominator here is going to be X plus seven times X minus seven. So we're going to multiply all three of those terms by X plus seven X minus seven. So now let's see what does that look like? We've got a nine X. And that's times X plus seven X minus seven. All over X plus seven plus we've got 63 times X plus seven X minus seven. That is all over X minus seven equals nine X squared plus times X plus seven X minus seven. And that is all over. And I'm going to write it as X plus seven X minus seven. Factored out instead of x squared minus 49. Great. Now is the fun part. We get to cancel out factors. So with that first term there is an X plus seven in both the numerator and the denominator. The second one we've got an x minus seven in the numerator and the denominator. And in this right hand side of the equal sign, there's an X plus seven in both and an X minus seven in both. The numerator and the denominator. So what have we got now Now we've got nine X times X minus seven plus 63 times X plus seven equals nine X squared plus 450. Yay that looks much better now we can go ahead and do our distribution here to get rid of our parentheses. So nine X times X. Is nine X squared nine X times negative seven is minus 63 X 63 times x. is plus 63 x 63 times seven is 441. Bring down the right hand side equals nine X squared plus 450. Okay on the left hand side looks like we've got some like terms here negative 63 X plus 63 X. Well those are going to cancel out. So now we've got nine X squared plus 441 equals nine X squared plus 450. Well this is starting to look a little bit nonsensical if we took and we wanted to bring the excess together we subtracted nine X squared from both sides. Well it's going to cancel out on both sides. And then all we've got is equals 450 that's that's nonsense. That's inconsistent. We look at our answer choices and we choose our answer choice. C Well done. We'll catch you on the next one

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The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x^2 + 36)/(x^2 - 9)