# 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 4𝑥 + √2 = 9√2.

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### Video Transcript

Find, in the set of real numbers, the solution set of the equation four 𝑥 plus root two equals nine root two.

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

In an equation, both sides of the equation are equal. In this case, we know that four 𝑥 plus root two is equal to nine root two. If we apply the same operations to both sides of the equation, then they will remain equal. We can also use this idea to isolate 𝑥 on one side of the equation and thus find the values of 𝑥 that satisfy the equation.

We can start by subtracting the square root of two from both sides of the equation. On the left-hand side of the equation, we know that root two minus root two is zero, so we are left with four 𝑥. On the right-hand side of the equation, we obtain nine root two minus root two. We can simplify the right-hand side of the equation by taking out the shared factor of root two from the two terms. This gives us nine minus one all multiplied by root two, which we can evaluate is equal to eight root two.

To isolate 𝑥 on the left-hand side of the equation, we need to divide both sides of the equation by four. On the left-hand side of the equation, we note that four over four is one. On the right-hand side of the equation, we have eight root two over four. We can then calculate that eight over four is two, leaving us with the equation 𝑥 equals two root two.

We cannot stop here. Remember that we are asked to find the solution set of the equation over the set of real numbers. So we need to check that our solutions are real numbers and give our answer as a set. Of course, we know that two root 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 of two root two.