Video Transcript
If 𝑥 is a solution to the equation
𝑥 plus one all squared is equal to eight, determine if 𝑥 is a member of the set of
rational numbers or 𝑥 is an element of the set of irrational numbers.
In this question, we are told that
𝑥 is a solution to a given equation. We need to use this fact to
determine if 𝑥 must be a rational or irrational number.
To answer this question, let’s
start by recalling what it means to say that a number is rational or irrational. First, we recall that ℚ is the set
of rational numbers. It is the set of all quotients of
integers such that the denominator is nonzero. If we take the complement of this
set, then we get the set of irrational numbers. This includes all numbers that
cannot be written as the quotient of two integers.
Therefore, to answer this question,
we need to determine if the solutions to the equation can be written as the quotient
of integers. Let’s find the solutions of the
equation. We can start by taking square roots
of both sides of the equation. Remember, we obtain a positive and
a negative root. We have that 𝑥 plus one is equal
to positive or negative root eight. We can then subtract one from both
sides of the equation to see that 𝑥 is equal to negative one plus or minus the
square root of eight. So there are two solutions to this
equation.
We need to determine if these
solutions are rational or irrational. And there are a few ways that we
can do this. One way is to recall that the
square root of a nonperfect square is irrational. Therefore, since eight is not a
perfect square, we can conclude that the square root of eight is an irrational
number.
However, this is not yet enough
information to answer the question. We can recall that all rational
numbers either have a finite or repeating decimal expansion. In fact, the reverse is also
true. If a number has a finite or
repeating expansion, then it is a rational number. We can think about the set of
irrational numbers in a similar way. These are the numbers with an
infinite nonrepeating decimal expansion. In the same way, the reverse is
true; any number that has an infinite nonrepeating decimal expansion is
irrational.
Let’s now consider the decimal
expansion of both solutions to this equation. First, we know that root eight has
an infinite nonrepeating decimal expansion since it is irrational. We do not need to know its actual
expansion. However, we will write this down as
2.82 and so on. Multiplying this decimal by
negative one will not change the fact that its decimal expansion is infinite and
does not repeat. So we can say that positive and
negative eight are both irrational.
Finally, we can consider what
happens to the decimal expansion of each of these numbers when we subtract one. We know that subtracting one from
these numbers will only change its unit digit, since this digit is nonzero. The decimal part of the expansion
will remain unchanged. Therefore, the decimal expansion of
these two solutions are both infinite and nonrepeating, since the decimal part of
these solutions are the same as positive and negative root eight. Hence, the answer is that 𝑥 must
be an irrational number.
This question highlights a useful
property of the sum of a rational and irrational number. In general, if 𝑎 is a rational
number and 𝑏 is an irrational number, then 𝑎 plus 𝑏 will always be an irrational
number. We can also use this property to
directly answer the question that 𝑥 must be an irrational number since it is the
sum of a rational and irrational number.
