Video Transcript
Find, in the set of real numbers, the solution set of the equation two minus root two 𝑥 is equal to three root two.
In this question, we are given a linear equation in a single variable and asked to find its solution set over the set of real numbers, which we can recall is the set of all real values that satisfy the equation. This means that we need to find all of the values of 𝑥 that satisfy the equation. To do this, we want to isolate 𝑥 on one side of the equation by applying the same operation to both sides of the equation.
Let’s start by subtracting two from both sides of the equation. On the left-hand side, we have that two minus two is zero. So, we obtain negative root two 𝑥 is equal to three root two minus two. We can then isolate 𝑥 on the left-hand side of the equation by dividing both sides of the equation through by negative root two. On the left-hand side of the equation, we have negative root two over negative root two is one. So, we are left with 𝑥 is equal to three root two minus two all divided by negative root two.
We want to give our value of 𝑥 in a simplified form, so we will rationalize the denominator. We can do this by multiplying the numerator and denominator by negative root two. This will not affect the value since this is the same as multiplying by one. We know that root two times root two is equal to two. So, we can evaluate the multiplication and simplify to get negative six plus two root two all over two. We can then cancel the shared factor of two in the numerator and denominator to get root two minus three. This is the only real solution to the equation.
Remember, we want to find the solution set of the equation. So, we need to write our answer as a set. Hence, the solution set of the given equation over the set of real numbers is the set containing only root two minus three.
