Explain the mistake that is made. Solve the equation (1)/(x)+(1)/(x-1)=(1)/(x(x-1)) Solution: Si

step 1 So in this problem, what we're being asked to do is to find out what mistake was made. So let's

Explain the mistake that is made. Solve the equation...

Explain the mistake that is made. Solve the equation $\frac{1}{x}+\frac{1}{x-1}=\frac{1}{x(x-1)}$ Solution: Simplify. $$ \begin{aligned} \frac{x(x-1)}{x}+\frac{x(x-1)}{x-1} &=\frac{x(x-1)}{x(x-1)} \\ (x-1)+x &=1 \\ x-1+x &=1 \\ 2 x &=2 \\ x &=1 \end{aligned} $$ This is incorrect. What mistake was made?

Explain the mistake that is made.
Solve the equation $\frac{1}{x}+\frac{1}{x-1}=\frac{1}{x(x-1)}$
Solution:
Simplify.
$$
\begin{aligned}
\frac{x(x-1)}{x}+\frac{x(x-1)}{x-1} &=\frac{x(x-1)}{x(x-1)} \\
(x-1)+x &=1 \\
x-1+x &=1 \\
2 x &=2 \\
x &=1
\end{aligned}
$$
This is incorrect. What mistake was made?

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Key Concepts

Domain Restrictions

When dealing with rational expressions, it is crucial to identify the values for which the denominator is zero because these values are excluded from the domain of the expression. Neglecting this step can lead to including solutions in the final answer that do not satisfy the original equation, as the expression is undefined at these points.

Clearing Denominators

Multiplying both sides of an equation by the least common denominator is a standard strategy to eliminate fractions and simplify the equation. However, this step must be done with caution because if the common denominator contains variable expressions, it may be zero for certain values, and multiplying by it with the assumption that it is nonzero may introduce errors or extraneous solutions.

Extraneous Solutions

Extraneous solutions are results that emerge from the algebraic manipulation of an equation but do not represent valid answers when substituted back into the original equation. In rational equations, these often occur when multiplying both sides by an expression that may be zero, thereby possibly masking restrictions on the variable and producing invalid results.

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