# Question Video: Finding the Solution Set to an Equation over the Set of Real Numbers Mathematics

Find, in ℝ, the solution set of the equation 𝑥 − 3√3 = √3.

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### Video Transcript

Find, in the set of real numbers, the solution set of the equation 𝑥 minus three root three is equal to root three.

In this question, we are given an equation in a variable 𝑥 and asked to find the solution set of the equation over the set of real numbers. To do this, we can start by recalling that the solution set of an equation over a set is the set of all values that satisfy the equation in that set. So, in this case, we want to find the set of all real values of 𝑥 that satisfy the given equation.

In an equation, both sides of the equation must be the same. This means that we can apply the same operations to both sides of the equation to leave them equal. We can use this to solve for 𝑥 by isolating 𝑥 on one side of the equation. To isolate 𝑥 on the left-hand side of the equation, we need to remove the constant value of negative three root three. We can do this by adding three root three. We need to add this value to both sides of the equation to make sure that they are still equal.

On the left-hand side of the equation, we have negative three root three plus three root three is zero, so we are left with 𝑥. On the right-hand side, we have root three plus three root three. We can simplify the right-hand side of the equation by noting that both terms share a factor of root three. Taking this factor out yields one plus three all multiplied by root three, which is equal to four root three.

We have shown that the only solution to the equation is 𝑥 equals four root three. However, remember that we are asked to find the solution set of the equation over the real numbers. This means that we need to check all of the solutions are real numbers and give our answer as a set. Of course, we know that four root three is a real number. And so we have shown that the solution set of the given equation over the set of real numbers is the set containing only the value of four root three.