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.
