1. **State the problem:** We want to simplify and understand the expression $\left|1+\frac{1}{x}\right|$. 2. **Recall the absolute value definition:** The absolute value $|a|$ of a number $a$ is the distance from zero on the number line, so it is always non-negative. For any real number $a$, $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$. 3. **Analyze the expression inside the absolute value:** $$1 + \frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x+1}{x}$$ 4. **Rewrite the absolute value:** $$\left|1 + \frac{1}{x}\right| = \left|\frac{x+1}{x}\right| = \frac{|x+1|}{|x|}$$ 5. **Interpretation:** The absolute value of a fraction is the fraction of the absolute values of numerator and denominator. 6. **Final simplified form:** $$\boxed{\frac{|x+1|}{|x|}}$$ This expression is defined for all $x \neq 0$ because division by zero is undefined.