If R is the set of real numbers, Q is the set of rational numbers, I is the set of integers, N is the set of natural numbers, and S is the set of irrational numbers, list the set(s)for which a solution to the given equation exists. If a solution to the given equation does not belong to any set, state none.
-$$0 . \overline { 7 } x = 8$$
A) I, Q, R
B) $N , I , Q , R$
C) $S , R$
D) $Q , R$

Question

Quizplus · Accepted Answer

To solve the equation $0.\overline{7}x = 8$, we first recognize that $0.\overline{7}$ is a rational number (it can be expressed as $\frac{7}{9}$). The equation can be rewritten as $\frac{7}{9}x = 8$. Solving for $x$ gives $x = \frac{8 \times 9}{7} = \frac{72}{7}$, which is a rational number because it can be expressed as a fraction of two integers. Since $x$ is rational, it also belongs to the set of real numbers. Therefore, the correct sets are Q (rational numbers) and R (real numbers).

If R Is the Set of Real Numbers, Q Is 0.7‾x=80 . \overline { 7 } x = 80.7x=8

If R Is the Set of Real Numbers, Q Is $0 . \overline { 7 } x = 8$