Find the solution set of the
equation 𝑥 divided by four plus eight equals negative two, in the natural
Our possible answers are: A) the
set of negative forty, B) an empty set called no solution, C) the set including the
number twenty-four, and D) the set including the number negative ten.
Before we begin, let’s talk about
what the natural numbers are. Natural numbers are your positive
counting numbers. So numbers such as one, two, three,
four, five. So when we solve our equation, 𝑥
divided by four plus eight equals negative two, our answer needs to belong in that
set of natural numbers. Essentially, it needs to be
In order to solve for 𝑥, we must
first subtract eight from both sides of the equation. The eights on the left-hand side
cancel, and on the right-hand side of the equation, negative two minus eight is
negative ten. Now I must get rid of the four. And to do so, we must multiply both
sides of the equation by four. This way, on the left-hand side,
the fours cancel. And on the right-hand side, we take
negative ten times four which is negative forty. So our final answer is: 𝑥 equals
negative forty. But there is an issue. 𝑥 equals negative forty does not
fall in the natural numbers.
Since our answer is negative, our
solution set is empty. The number negative forty does not
belong in the set of natural numbers. So again, the solution set is
empty, written as the no solution. So our answer is B.
When we began this problem, we
actually could’ve eliminated two of our options. Since we knew our solution set
needed to be in the natural numbers, we couldn’t have a negative answer. So we could’ve crossed out A,
negative forty, and D, negative ten, leaving us only two options to choose from. But using the same calculations in
work, we would still find our answer to be B, no solution. It’s an empty set.