Video Transcript
Find, in the set of real numbers,
the solution set of the equation three root three π₯ plus one is equal to root three
π₯ plus four.
In this question, weβre given an
equation in a variable π₯ and asked to find the solution set of this equation in the
set of real numbers.
To do this, we can start by
recalling that the solution set of the equation in the set of real numbers means the
set of all real values that satisfy the given equation. Since both sides of the equation
are equal, we can apply the same operations to both sides of the equation and they
will remain equal. We can use this to isolate π₯ on
one side of the equation.
We can begin by subtracting one
from both sides of the equation. On the left-hand side of the
equation, we have one minus one is zero. And on the right-hand side of the
equation, we have four minus one is three. So, we obtain three root three π₯
equals root three π₯ plus three.
Since we want to isolate π₯ on one
side of the equation, we now need to subtract root three π₯ from both sides of the
equation. This gives us that three root three
π₯ minus root three π₯ is equal to three. To isolate π₯ on the left-hand side
of the equation, we can take out the shared factor of π₯ from both terms. This gives us that three root three
minus root three times π₯ equals three.
We can then note that three root
three minus root three is equal to two root three. This gives us the equation two root
three π₯ equals three. To isolate the variable π₯ on the
left-hand side of the equation, we need to divide both sides of the equation by two
root three. This gives us that π₯ must be equal
to three divided by two root three. This is the only solution to this
equation; however, we should rationalize the denominator to write our answer in a
standard form.
We rationalize the denominator by
multiplying both the numerator and denominator by the square root of three. This is equivalent to multiplying
by one, so it does not affect the solution. In the denominator, we can note
that root three times root three is three. So, we have three root three over
two times three. We can then cancel the shared
factor of three in the numerator and denominator to see that the only solution is π₯
equals root three over two.
Remember though we are asked to
find the solution set of the equation over the set of real numbers. So, we need to give our answer as a
set and check that the answers are real numbers. In this case, we know that root
three over two is a real number. So, the solution set of the given
equation over the set of real numbers is the set containing only the value root
three over two.
