Video: Subtracting Algebraic Fractions with 𝑥 in the Denomerator

Simplify (3/(3 − 𝑥)) − (6/(6 − 𝑥)).


Video Transcript

We’re going to simplify three over three minus 𝑥 minus six over six minus 𝑥, so let’s do what we did before. And that is to put brackets around the denominators and then we’ve got to find equivalent fractions, so that we end up with a common denominator. So we’re gonna take this three minus 𝑥 and multiply the second fraction top and bottom by that value. And we’re gonna take the six minus 𝑥 and we’re gonna multiply the top and bottom of the first fraction by that value. So we’ve got a three lots of six minus 𝑥 as the numerator for the first fraction and six lots of three minus 𝑥 for the second numerator. So we’ll bring those terms together to make one single fraction. This is what we’ve got. and we’ve gotta be very very careful when we multiply out these brackets, because that second term on the numerator is minus the whole thing. So let’s have a look at how that works. So we’re gonna have three lots of six and three lots of negative 𝑥. Now three lots of six is eighteen and three lots of negative 𝑥 is negative three 𝑥, but now we are taking away six lots of three. And we’re taking away six lots of negative 𝑥, so six lots of negative 𝑥 is negative six 𝑥. If we’re taking away negative six 𝑥, that means that we’re adding six 𝑥. Now looking at that numerator, we’ve got eighteen take away eighteen, which is nothing. And we’ve got negative three 𝑥 add six 𝑥, which is positive three 𝑥. And none of those terms will cancel down, so we’re left with our answer of three 𝑥 all over six minus 𝑥 times three minus 𝑥.

So the thing really to pay attention to there was this step here. When we were dealing with taking away things. Remember we had to take away the whole of this expression here, which meant being very careful about the signs that we ended up with here. That’s the most common place that people go wrong when they do these questions. So really do watch out for that.

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