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(male narrator) In this video,
we're gonna look at solving rational equations
by clearing the denominator.
You may recall solving problems
with fractions and variables in them before.
We would find the least common denominator--
in this case, it would be 12-- and we multiply each fraction--
or each term would be a better way to say it--by 12.
We could then reduce the 12 over 4 to 3,
the 12 over 2 to 6, and the 12 over 6 to 2.
And multiply the remaining stuff together:
3 times 3x would be 9x, minus 6 times 1 is 6,
equals 5 times 2 is 10.
And we would end up with an equation we could solve.
As our rational expressions become more complex,
we will use this same exact strategy.
We will clear fractions by multiplying...
by the least common denominator.
Let's take a look at some examples
where we use this strategy to clear the fractions.
In this problem, finding our least common denominator,
we see the number 7 and the variable x in the denominators.
We will multiply each term by the 7x.
This includes the minus 4, which is not a fraction.
When we do this,
we can go through and divide out the x from the first fraction,
the 7 and the x from the next fraction,
and what we're left with is: 7 times 5, or 35; equals 3;
minus 4; times 7x, which is 28x.
We can now solve this problem for x like always
by subtracting 3 to get 32, equals -28x.
And then finally dividing both sides by -28
to get our solution.
When we reduce, dividing by 4, we get -8/7.
Let's try one more example
where we have to clear the denominator
by multiplying by the least common denominator.
In this problem,
our denominators are a binomial: x plus 5.
This means my least common denominator
is going to be that binomial: x plus 5.
What we'll do with this
is we will multiply each term by x plus 5.
Notice this includes the x, which was not a fraction.
When we do this,
the x plus 5s will divide out in the fractions,
and we're just left
with 4 plus x, times x, plus 5, equals -2.
We now have an equation we can solve
by first distributing the x:
4 plus x squared, plus 5x, equals -2.
And then, we can make the equation equal to 0,
so we can decide how to solve-- by adding 2 to both sides.
Putting this in order, where the x squared comes first,
we get x squared, plus 5x, plus 6, equals 0.
From here, we could either
complete the square or use the quadratic formula,
but it might be easier if this equation factors.
Sure enough, it does factor quite nicely:
x plus 2, times x, plus 3.
We can now set each factor equal to 0:
x plus 2, equals 0; and x plus 3, equals 0;
and solve those equations to get our x.
Subtracting 2 tells us
that x is equal to -2.
Or subtracting 3 tells us
that x is equal to -3.
These equations made up of ratios are easily solved
by first clearing the denominator
and multiplying by the least common denominator.