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All courses Math III · A-REI.2 13 of 55
Solve simple rational and radical equations and identify extraneous solutions.

Solve a rational equation with one denominator by clearing the denominator and checking restrictions

Problem
Solve this rational equation with one denominator: (x+3)/(x-1)=5
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Hint

State the denominator restriction first, then clear the denominator by multiplying both sides by \(x-1\).

In a rational equation, denominator values that make the expression undefined must be excluded even after solving.

Solution walkthrough

01

Write the restriction

\[x\ne~1\]

The denominator \(x-1\) cannot equal \(0\).

02

Clear the denominator

\[\frac{x+3}{x-1}~=~5\ \Rightarrow\ x+3~=~5(x-1)\]

Multiply both sides by \(x-1\) to remove the fraction.

03

Solve the linear equation

\[x+3~=~5x-5\ \Rightarrow\ 8~=~4x\ \Rightarrow\ x~=~2\]

Collect like terms and solve for \(x\).

04

Check against the restriction

\[2\ne~1\]

The solution does not violate the denominator restriction, so it is valid.

05

State the final answer

\[\text{The}~\text{solution}~\text{is}~x~=~2,~\text{with}~x≠1.\]

The solution is \(x=2\), with \(x≠1\).

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Another way

  1. You can think of clearing the denominator as cross-multiplying the equation by the denominator factor \(x-1\).

!

Common mistake

A common mistake is to solve for \(x\) but forget to include or check the restriction \(x≠1\).