Question Video: Finding the Solution Set to a Linear Equation That Includes a Radical and Representing It on a Number Line Mathematics

Find, in ℝ, the solution set of the equation √3𝑥 + 2 = 5. Which of the following shows the solution of the equation on a 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 root three 𝑥 plus two is equal to five. Which of the following shows the solution of the equation on a number line? Is it option (A), option (B), option (C), option (D), or option (E)?

The first part of this question involves solving a linear equation of the form 𝑎𝑥 plus 𝑏 equals 𝑐, where 𝑎, 𝑏, and 𝑐 are nonzero constants. To solve the equation root three 𝑥 plus two is equal to five, we begin by isolating the 𝑥-term. We do this by subtracting two from both sides. Root three 𝑥 is therefore equal to three. Next, we divide through by root three. On the left-hand side, the root threes cancel and we are left with 𝑥. Our expression on the right-hand side, three over root three, is a fraction, and its denominator is irrational. We therefore need to rationalize the denominator by multiplying both the numerator and denominator by root three. And since root three multiplied by root three is three, we have 𝑥 is equal to three root three over three. And this simplifies to 𝑥 is equal to root three.

The solution set of the equation root three 𝑥 plus two equals five is the single value root three.

The second part of this question asks us to identify the number line that represents this solution. Since the square root of one is equal to one and the square root of four is equal to two, we know that the square root of three must lie between one and two. This means that we can rule out options (B), (C), (D), and (E), as (B) has a solution between three and four, (C) and (D) have a solution between zero and one, and (E) has solution 𝑥 is equal to three. The correct number line is therefore option (A). Whilst it is not required in this question, it is worth noting that the square root of three is equal to 1.732 and so on. And since the solid dot lies closer to two than one, this backs up the answer of option (A).

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