Question Video: Solving an Equation and Finding the Solution on a Number Line Mathematics

Find in ℝ the solution set of the equation 𝑥 + 2√3 = 3√3. Which of the following represents the solution of the equation on the number line? [A] Option A [B] Option B [C] Option C [D] Option D [E] Option E

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Video Transcript

Find in the set of real numbers the solution set of the equation 𝑥 plus two root three is equal to three root three. Which of the following represents the solution of the equation on the number line?

In this question, we are given an equation in an unknown 𝑥 and asked to find the solution set of the equation over the set of real numbers. We then need to determine which of five given number lines correctly represents the solution set.

To answer this question, we can begin by recalling that the solution set to an equation over the set of real numbers is the set of all real values of 𝑥 that satisfy the equation. To solve our equation for 𝑥, we want to isolate 𝑥 on one side of the equation. We can do this by subtracting two root three from both sides of the equation. This then gives us that 𝑥 is equal to three root three minus two root three.

We can then evaluate the right-hand side of the equation by taking out the shared factor of root three to obtain three minus two multiplied by the square root of three, which we can then calculate is equal to the square root of three. Remember, we want to find the solution set of the equation. So, we need to write our answer as a set. We have the set only containing the value of root three.

In the second part of this question, we want to correctly identify the square root of three on a number line. The first thing we should do is determine which integers root three lies between. We can bound root three between two integers by first noting that root three squared is three. This is bigger than one squared since that is just equal to one. However, it is smaller than two squared, which is four.

This is useful because we know that if we have three nonnegative numbers 𝑎, 𝑏, and 𝑐, then we can compare the sizes of these numbers by comparing the sizes of their squares. In general, for nonnegative values 𝑎, 𝑏, and 𝑐, if 𝑎 squared is less than 𝑏 squared is less than 𝑐 squared, then 𝑎 is less than 𝑏 is less than 𝑐.

In our case, we have 𝑎 equals one, 𝑏 equals root three, and 𝑐 equals two. So, the square root of three must lie between one and two, which we can see is the number line in option (E).

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