Video: Subtraction of Rational Expressions Where the Denominators are Equal

Simplify (4𝑥 − 2)/(2𝑥² − 1) − (3𝑥 + 8)/(2𝑥² − 1).

02:40

Video Transcript

Simplify four 𝑥 minus two over two 𝑥 squared minus one minus three 𝑥 plus eight over two 𝑥 squared minus one.

Okay, so what we have here are two algebraic fractions. And we’re actually subtracting one from the other. The key thing here is they both have the same denominator, because this is going to make life a lot easier. And actually when we’re doing this, we do it in exactly the same way that we’d actually subtract any fractions that have the same denominator.

So I’ve got this example: seven-eighths minus five-eighths. Well as they actually have the same denominator, we can actually rewrite our fraction like this: so seven minus five over eight, which gives us the answer two over eight or two-eighths. So as you can see here, all we’ve done is actually subtracted the numerators from each other.

So because we have the same denominator, we’ll do that with our expression. So we can say that four 𝑥 minus two over two 𝑥 squared minus one minus three 𝑥 plus eight over two 𝑥 squared minus one is equal to four 𝑥 minus two minus three 𝑥 plus eight all over two 𝑥 squared minus one. I’ve actually put the parentheses into this question because of this negative sign.

And I wanted to put them in just because when we’re actually going to subtract the values from each other, this will affect the signs. So as you can see, we get four 𝑥 minus two minus three 𝑥, that’s because we had the negative in front of the parentheses, and then minus eight. This is the bit where people make the most common error. And it’s because they’ll often put plus eight on the end.

But in fact, we had a negative in front of the parentheses. And so if we’re minus-ing a positive, then it’s going to be a minus or a negative. Great! Okay, so now all we need to do is actually simplify the numerator. And when we do that, we get four 𝑥 minus three 𝑥, which is just 𝑥, and then negative two minus eight, which is negative 10.

So we’ve got 𝑥 minus 10 over two 𝑥 squared minus one. So we can say that if we simplify four 𝑥 minus two over two 𝑥 squared minus one minus three 𝑥 plus eight over two 𝑥 squared minus one, we get 𝑥 minus 10 over two 𝑥 squared minus one. But the key thing to remember with this question is actually if you ever get a question like this and it looks sort of more complicated, you got algebraic fractions, do remember the basics. And remember these basic rules, like I said with when we’ve got the same denominator, we can just add or subtract the numerators.

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