# Question Video: Finding a Solution Set over the Real Numbers Mathematics

Find, in ℝ, the solution set of the equation 2(3𝑥 + √5) = 8√5. Which of the following shows the solution set of the equation on a number line? [A] Option A [B] Option B [C] Option C [D] Option D

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### Video Transcript

Find, in the set of real numbers, the solution set of the equation two times three 𝑥 plus the square root of five is equal to eight root five. Which of the following shows the solution set of the equation on a number line?

In this question, we are given an equation involving an unknown 𝑥. And we need to determine the solution set of the equation over the set of real numbers and then identify this set on a number line.

To do this, we first recall that the solution set means the set of all real values that satisfy the equation. We can solve for the values of 𝑥 by isolating 𝑥 on one side of the equation. We can start by dividing both sides of the equation by two. On the left-hand side of the equation, we have two over two, which is equal to one. And on the right-hand side of the equation, we have eight over two equals four. So we have three 𝑥 plus root five equals four root five.

Now, we want to subtract the square root of five from both sides of the equation. On the left-hand side of the equation, we have root five minus root five is zero. And on the right-hand side of the equation, we have four root five minus root five is three root five. So we are left with three 𝑥 equals three root five.

We can now isolate 𝑥 on the left-hand side of the equation by dividing both sides of the equation by three. On both sides of the equation, we have three over three is one. So we are left with 𝑥 is equal to the square root of five. Remember, we want the solution set of the equation, and we know that root five is a real number. So we have the set containing only the value of root five.

For the second part of this equation, we need to identify which number line correctly shows the solution of the equation, that is, the square root of five. We can calculate that two squared is four, root five squared is five, and three squared is nine. So root five lies between two and three. We can see that only the number line in option (A) has the point between two and three. So it must be the correct representation of the square root of five.