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# Equations with rational expressions

Sal solves (x²-10x+21)/(3x-12)=(x-5)/(x-4), which has one real solution and one extraneous solution.

## Want to join the conversation?

• Why can't the denominator be zero?
• Division by zero is an undefined operation.

Let 𝑦 = 𝑥∕0 ⇒ 𝑥 = 0 ∙ 𝑦
For 𝑥 ≠ 0, there is no finite value 𝑦 that satisfies these equations, so 𝑦 is undefined.
For 𝑥 = 0, any finite value 𝑦 satisfies the equations, so again 𝑦 is undefined.
• Near the end of the video, Sal says that x cannot equal 4 because it would have both sides undefined. But wouldn't 4 actually work? Think about it: if x is 4, the left side has division by zero (which is undefined), and the right side has division by zero (which is undefined). If both sides are undefined, that means that both sides are equal, because "undefined" is equal to "undefined". So shouldn't 4 technically be a solution?
• When something is "undefined" it has no answer. It doesn't matter that the 2 sides are equal. You can't find a numeric result when you have an undefined condition.
• why is this so complicated?
• can any letter be a variable? I have been wondering this forever!
• Any symbol at all can be a variable, if you can write it consistently. Latin, Greek, and Hebrew letters all appear as variables in some parts of math. You can use any of those, draw your own symbols, use emoji, or whatever you like. They're all valid variables.
• So my teacher has been teaching this, and he says we have to find the LCD. Is that what is happening in this video?
• No. The LCD is the Least Common Denominator. Sal found the answer in a different form. There is a video on this, but I'm not sure how to link it.
• why cant we just multiply the denominator right away before simplifying it
• You can, but simplifying makes it easier to understand.
• Can the numerator equal 0? I know the denominator can’t equal 0 but can the numerator?
• Yes, its ok for the numerator to = 0. The one exception to this is if you are dividing by a fraction with a numerator of 0. Once it flips, you would have a denominator of 0 which would not be good.
Hope this makes sense.
• how is the multiplying both sides by x-4 working?
seriously this first step is quite confusing.
(1 vote)
• You're multiplying the same thing on both sides, so equality isn't disturbed.

Imagine having 10 = 10. If I multiply 5 on both sides, I get 50 = 50, which is true. So, similarly, multiplying (x-4) is perfectly valid
• what happens when the denominator looks like (x+4)(x-1) or something like that?
• If the denominator were to have two binomials like that, do the same thing as you would in the video. Expectedly, it's a little bit more annoying as now as you have a third degree polynomial, but otherwise the same procedure as in the video.

Hopefully that helps !
• Is there no easy way to solve this via cross multiplication? Cross multiplying I get:
x^3-14x^2+61x-84 = 3x^2-27x+60

subtracting both sides:
x^3-17x^2+88x=144
and I'm stuck here.
(1 vote)
• The problem with cross multiplication is that you end up with a much more difficult problem. Your polynomial takes a lot more work to try and find the factors.

The technique shown in the video eliminates some factors in the denominators making the problem more manageable and simple to solve.

Use cross multiplication when the denominators are simple monomials. If they are binomials or larger, then use the method in this video.