A rational equation is an equation that contains one or more rational expressions.
A rational expression is a fraction whose numerator, denominator, or both contain variables.
The key rule is that denominators cannot be zero.
1. What is a rational equation?
Examples of rational equations include:
\[
\begin{aligned}
\frac{x+2}{x-3}&=\frac{7}{2},\\
\frac{1}{x-2}+\frac{1}{x+2}&=1,\\
\frac{x^2-1}{x-1}&=x+1.
\end{aligned}
\]
The variable appears in at least one denominator, so the first step is always to find excluded values.
2. Domain restrictions
A rational expression is undefined whenever its denominator is zero.
Therefore, for each denominator \(D(x)\), we require:
\[
\begin{aligned}
D(x)&\ne 0.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\frac{x+2}{x-3}
\end{aligned}
\]
is undefined when:
\[
\begin{aligned}
x-3&=0,\\
x&=3.
\end{aligned}
\]
So the domain restriction is:
\[
\begin{aligned}
x&\ne 3.
\end{aligned}
\]
3. Clearing denominators
To solve a rational equation, multiply both sides by a common denominator.
This removes fractions and produces a polynomial equation.
For example:
\[
\begin{aligned}
\frac{x+2}{x-3}&=\frac{7}{2}.
\end{aligned}
\]
The denominators are \(x-3\) and \(2\), so a common denominator is \(2(x-3)\).
\[
\begin{aligned}
2(x-3)\cdot\frac{x+2}{x-3}
&=
2(x-3)\cdot\frac{7}{2}.
\end{aligned}
\]
Cancel common factors:
\[
\begin{aligned}
2(x+2)&=7(x-3).
\end{aligned}
\]
4. Worked example
Solve:
\[
\begin{aligned}
\frac{x+2}{x-3}&=\frac{7}{2}.
\end{aligned}
\]
Domain restriction:
\[
\begin{aligned}
x-3&\ne0,\\
x&\ne3.
\end{aligned}
\]
Clear denominators:
\[
\begin{aligned}
2(x+2)&=7(x-3).
\end{aligned}
\]
Expand and solve:
\[
\begin{aligned}
2x+4&=7x-21,\\
25&=5x,\\
x&=5.
\end{aligned}
\]
Check the restriction:
\[
\begin{aligned}
5&\ne3.
\end{aligned}
\]
Verify in the original equation:
\[
\begin{aligned}
\frac{5+2}{5-3}
&=
\frac{7}{2}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=5}.
\end{aligned}
\]
5. Why excluded values matter
Clearing denominators can create candidate values that were not allowed in the original equation.
For example:
\[
\begin{aligned}
\frac{x}{x-1}&=1+\frac{1}{x-1}.
\end{aligned}
\]
The denominator gives:
\[
\begin{aligned}
x-1&\ne0,\\
x&\ne1.
\end{aligned}
\]
If a later algebraic step produces \(x=1\), it must be rejected because the original equation is undefined at \(x=1\).
6. Holes and removable factors
A rational expression can simplify algebraically while still having an excluded value.
For example:
\[
\begin{aligned}
\frac{x^2-1}{x-1}
&=
\frac{(x-1)(x+1)}{x-1}.
\end{aligned}
\]
For \(x\ne1\), this simplifies to \(x+1\). But \(x=1\) is still excluded from the original expression.
This creates a hole in the graph.
7. Graph interpretation
A rational equation:
\[
\begin{aligned}
\text{LHS}(x)&=\text{RHS}(x)
\end{aligned}
\]
can be graphed as two curves:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Verified real solutions appear where the two graphs intersect and all denominators are nonzero.
Excluded values often appear as vertical asymptotes or holes.
8. Common-denominator method
If the equation is:
\[
\begin{aligned}
\frac{A(x)}{B(x)}&=\frac{C(x)}{D(x)},
\end{aligned}
\]
then multiply both sides by \(B(x)D(x)\):
\[
\begin{aligned}
A(x)D(x)&=C(x)B(x).
\end{aligned}
\]
Then solve:
\[
\begin{aligned}
A(x)D(x)-C(x)B(x)&=0.
\end{aligned}
\]
But the restrictions remain:
\[
\begin{aligned}
B(x)&\ne0,\\
D(x)&\ne0.
\end{aligned}
\]
9. Formula summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
10. Common mistakes
- Solving first and checking denominator restrictions only at the end.
- Forgetting that canceled factors can still create excluded values.
- Multiplying only part of the equation by the common denominator.
- Dividing by an expression that might be zero.
- Reporting an excluded value as a solution.
- Assuming every intersection-looking point on a graph is valid without checking denominators.
Key idea: clear denominators to solve, but keep denominator restrictions from the original equation.
