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).