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### Course: Get ready for AP® Calculus>Unit 7

Lesson 1: Rational equations

# 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.
• 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.
• why is this so complicated?
• 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.
• why isn't four a solution?
i know that both it would mean dividing by zero, but then you would get on both sides of the equation
undefined = undefined, and thats true!
undefined does = undefined.
so doesn't four uphold the equality of the equation?
• Undefined doesn't equal undefined, it doesn't equal anything. 'Undefined' is not a special numeric value, it means 'this cluster of symbols is meaningless'. Saying that two undefined expressions are equal is like taking two random keysmashes and declaring one to be prime and the other composite.
• 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