If n is a rational number, 1/n does not equal n-1.Question 4Select one:TrueFalse

Analysis of the Statement

To determine whether the statement "If $n$ is a rational number, $\frac{1}{n} \neq n - 1$" is True or False, we can examine the equation.

1. Condition: $n$ is a rational number.
2. Statement to Analyze: Assess the relationship $\frac{1}{n}$ and $n - 1$.

Step 1: Solve the Inequality

To see if $\frac{1}{n}$ can equal $n - 1$:

$\frac{1}{n} = n - 1$

Step 2: Rearrange the Equation

Multiply both sides by $n$ (assuming $n \neq 0$ since $n = 0$ is not rational):

$1 = n(n - 1)$

Step 3: Expand and Rearrange

This gives:

$1 = n^2 - n$ $n^2 - n - 1 = 0$

Step 4: Apply the Quadratic Formula

Applying the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1, b = -1, c = -1$:

$n = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}$ $n = \frac{1 \pm \sqrt{1 + 4}}{2}$ $n = \frac{1 \pm \sqrt{5}}{2}$

Step 5: Evaluate the Roots

The two solutions will be:

$n_1 = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad n_2 = \frac{1 - \sqrt{5}}{2}$

Both roots are irrational numbers, which means that there are no rational values of $n$ for which $\frac{1}{n} = n - 1$.

Conclusion

Since the only values of $n$ that satisfy $\frac{1}{n} = n - 1$ are irrational, we can conclude that for any rational $n$, $\frac{1}{n} \neq n - 1$ holds true.

Final Answer

True