Question Video: Subtracting Algebraic Fractions with 𝑥 in the Denominator | Nagwa Question Video: Subtracting Algebraic Fractions with 𝑥 in the Denominator | Nagwa

Question Video: Subtracting Algebraic Fractions with 𝑥 in the Denominator Mathematics

Simplify (3/(3 − 𝑥)) − (6/(6 − 𝑥)).

03:02

Video Transcript

Simplify three over three minus 𝑥 minus six over six minus 𝑥.

In this question, we are asked to simplify the difference between two fractions. We can start by noting that the fractions contain an unknown 𝑥, so these are algebraic fractions. We treat algebraic fractions in the same way that we treat regular fractions. The first thing that we want to do is to find equivalent fractions with the same denominator so that we can find the difference between the fractions by subtracting their numerators. We can do this by multiplying both the numerator and denominator of the first fraction by six minus 𝑥 and multiplying the numerator and denominator of the second fraction by three minus 𝑥, where we note that we are not changing the value of the expression since in both cases we are multiplying by one.

To multiply two fractions, we multiply their numerators and denominators separately. Doing this and rearranging gives us three times six minus 𝑥 over three minus 𝑥 times six minus 𝑥 minus six times three minus 𝑥 over three minus 𝑥 times six minus 𝑥. The denominators of the two fractions are now the same, so we can evaluate this difference by subtracting the numerators. This gives us three times six minus 𝑥 minus six times three minus 𝑥 all over three minus 𝑥 times six minus 𝑥.

We now need to simplify this expression. We could take out the shared factor of three in the numerator or distribute both factors over the parentheses. We will start by distributing. This gives us 18 minus three 𝑥 minus 18 plus six 𝑥 all over three minus 𝑥 times six minus 𝑥. We can then calculate that 18 minus 18 is zero and negative three 𝑥 plus six 𝑥 is three 𝑥, leaving us with three 𝑥 over three minus 𝑥 times six minus 𝑥. We cannot cancel any shared factors in the numerator and denominator, so we cannot simplify any further.

Therefore, we were able to simplify three over three minus 𝑥 minus six over six minus 𝑥 to get three 𝑥 over three minus 𝑥 times six minus 𝑥.

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