Is the number 9 x –2 rational or irrational?

Question

GPT-4o mini · Answer

To determine if the expression $ 9x - 2 $ is rational or irrational, we need to analyze the nature of the variable $ x $. 1. **Rational Case**: If $ x $ is a rational number (i.e., it can be expressed as the fraction of two integers), then $ 9x $ is also rational because multiplying a rational number by an integer (9 in this case) yields a rational number. Since the subtraction of two rational numbers (like $ 9x $ and 2) results in a rational number, $ 9x - 2 $ would also be rational. 2. **Irrational Case**: If $ x $ is an irrational number (i.e., it cannot be expressed as a fraction of two integers), then $ 9x $ would be irrational as well. When we subtract a rational number (2) from an irrational number, the result $ 9x - 2 $ would still be irrational. In conclusion, whether $ 9x - 2 $ is rational or irrational depends on the value of $ x $: - If $ x $ is rational, $ 9x - 2 $ is rational. - If $ x $ is irrational, $ 9x - 2 $ is irrational. Therefore, without specific information about $ x $, we cannot definitively label $ 9x - 2 $ as rational or irrational; it depends on the value of $ x $.