Rational Equations Let You Solve Equations With Algebraic Fractions
- Rational equations are equations that include fractions whose numerators and/or denominators are algebraic expressions.
- They appear often in scientific and technical contexts where quantities combine "in parts" (for example, shared electrical current, combined rates, or inverse relationships).
- Solving them relies on the same logic you already use with numerical fractions, but with one extra layer: some values of the variable are not allowed.
Definition
Rational equation
An equation that includes one or more rational algebraic expressions.
Because denominators can become zero, rational equations often produce extraneous solutions (answers that come from algebraic steps but do not satisfy the original equation).
Definition
Extraneous solution
A value found while solving an equation algebraically that does not satisfy the original equation (often because it makes a denominator zero).
Domain Restrictions Must Be Checked First And Last
- Before doing any algebra, identify domain restrictions (values that make any denominator zero).
- These values can never be solutions.
Example
For example, in $$\frac{3}{x}=\frac{8}{x-2}$$
the restrictions are $x\neq 0$ and $x\neq 2$.
You should think of these restrictions as part of the problem statement: you are solving the equation only for allowable $x$ values.
Common Mistake
- A very common mistake is to solve, obtain answers, and stop.
- You must reject any answer that makes a denominator zero, and you should check remaining answers by substitution.
Two Equivalent Core Methods: Clear Denominators Or Match Denominators
- Most rational equations are solved using equivalence transformations (operations that keep the solution set the same, as long as you respect restrictions).
- Two closely related strategies dominate.
Method A: Multiply Through By The LCM (Clear Denominators)
This is often the fastest general approach.
- Factor each denominator completely.
- Find the least common multiple (LCM) of all denominators.
- Multiply every term on both sides by the LCM.
- Simplify to a polynomial equation (often linear or quadratic).
- Solve, then check in the original equation and reject extraneous solutions.
Method B: Rewrite Everything With A Common Denominator
This method makes the "fraction logic" very explicit.
- Factor denominators and determine the LCM.
- Rewrite each term so all fractions share the LCM as denominator.
- If you reach $\frac{A(x)}{D(x)}=\frac{B(x)}{D(x)}$ (with $D(x)\neq 0$), set numerators equal: $A(x)=B(x)$.
- Solve, then check.
Note
Method B can feel "cleaner" when the equation naturally becomes "same denominator on both sides," but Method A is often quicker when there are many terms.
Cross-Multiplication Works Only In A Specific Situation
- If your equation is exactly one fraction equals one fraction, you may cross-multiply: $$\frac{a}{b}=\frac{c}{d}\quad \Rightarrow\quad ad=bc,$$ provided $b\neq 0$ and $d\neq 0$.
- This is not a separate trick, it is just Method A applied with LCM $bd$.
Common Mistake
- Do not cross-multiply across addition or subtraction.
- For example, $$\frac{1}{x}+\frac{1}{x+1}=\frac{2}{x}$$ cannot be solved by multiplying "diagonally." You must use an LCM.
Example
Modelling With A Rational Equation
- Two appliances share a total electrical current of 16 amps, modelled by $$\frac{x}{10}+\frac{x}{30}=16,$$ where $x$ is the voltage (in volts) used by each appliance.
- The denominators are numbers, so there are no variable restrictions here.
- Clear denominators by multiplying by the LCM of 10 and 30, which is 30: $$30\left(\frac{x}{10}+\frac{x}{30}\right)=30\cdot 16.$$
- Simplify: $$3x+x=480\Rightarrow 4x=480\Rightarrow x=120.$$
- Check by substitution: $$\frac{120}{10}+\frac{120}{30}=12+4=16.$$
- So each appliance uses 120 V.
Exam technique
- When denominators are only numbers, LCM clearing is straightforward.
- When denominators involve $x$, always write restrictions first, then multiply.
Example question
Solve and check: $$\frac{1}{x-6}+\frac{x}{x-2}=\frac{4}{x^2-8x+12}$$
Solution
Step 1: Factor and restrict
- Factor the quadratic denominator: $$x^2-8x+12=(x-6)(x-2)$$
- Restrictions: $x\neq 6$ and $x\neq 2$.
Step 2: Multiply through by the LCM
- The LCM of $x-6$, $x-2$, and $(x-6)(x-2)$ is $(x-6)(x-2)$.
- Multiply both sides by $(x-6)(x-2)$: $$ (x-6)(x-2)\left(\frac{1}{x-6}+\frac{x}{x-2}\right)=(x-6)(x-2)\cdot \frac{4}{(x-6)(x-2)}$$
- Simplify term-by-term: $$ (x-2)+x(x-6)=4$$
- Expand and rearrange: $$x^2-5x-2=4\Rightarrow x^2-5x-6=0$$
- Factor: $$(x-6)(x+1)=0\Rightarrow x=6\text{ or }x=-1$$
Step 3: Reject extraneous and check
- $x=6$ is not allowed, so it is extraneous.
- Check $x=-1$:
- LHS: $$\frac{1}{-1-6}+\frac{-1}{-1-2}=-\frac{1}{7}+\frac{1}{3}=\frac{4}{21}$$
- RHS: $$\frac{4}{(-1)^2-8(-1)+12}=\frac{4}{1+8+12}=\frac{4}{21}$$
- So the solution is $x=-1$.
Why Matching Denominators Lets You Set Numerators Equal
- If you reach $$\frac{A(x)}{D(x)}=\frac{B(x)}{D(x)},$$ then for any $x$ where $D(x)\neq 0$, the equality holds exactly when $$A(x)=B(x)$$
- This generalizes the numerical fact: $$\frac{a}{b}=\frac{c}{b}\Rightarrow a=c\quad (b\neq 0)$$
- It also connects to cross-multiplication: from $\frac{a}{b}=\frac{c}{d}$, multiplying both sides by $bd$ gives $ad=bc$.
Note
You are allowed to set numerators equal only after confirming the common denominator is not zero for the values you keep.
Graphical Meaning: Solutions Are Intersections
- Rational equations can also be solved graphically.
- Graphing is useful to:
- estimate solutions before doing algebra,
- check whether you should expect 0, 1, or 2 solutions,
- spot suspicious answers near vertical asymptotes.
Example
- For example, define $$f(x)=\frac{1}{x-6}+\frac{x}{x-2},\qquad g(x)=\frac{4}{x^2-8x+12}$$
- The solutions to the original equation are the $x$-values where the graphs of $f$ and $g$ intersect, subject to the same restrictions ($x\neq 2,6$).
Common Mistake
- A graph may appear to "meet" near a vertical asymptote, but points where a function is undefined can never be solutions.
- Confirm solutions algebraically.
Modelling With Rational Equations: Interpret Restrictions In Context
- Many real situations create rational equations because of inverse relationships.
- Electrical models often include terms like $\frac{V}{R}$ (current contribution when voltage is $V$ and resistance is $R$).
- Rate problems often involve terms like $\frac{1}{t}$, and adding these terms represents combined work or flow rates.
- In modelling, restrictions such as $x\neq 0$ often have a real meaning (a time cannot be zero, a resistance cannot be zero in the model, a length cannot take a value that makes the formula undefined).
Case study
- Mains electricity varies by region, but the same rational relationships are used in circuit modelling.
- Solving rational equations lets engineers test whether proposed voltage and resistance values are consistent with a desired current.
Common Mistake
Here is the list of common pitfalls you should avoid:
- Not factoring first, leading to an incorrect LCM.
- Forgetting restrictions, so an invalid value is kept.
- Algebra errors when expanding after multiplying by the LCM.
- Misusing cross-multiplication, which only applies to "one fraction equals one fraction."
Tip
- After solving, check by substitution like this: compute LHS and RHS separately at your candidate $x$.
- If both are defined and equal, the solution is valid.
Self review
- In $\frac{10}{x+4}=\frac{15}{4(x+1)}$, what values of $x$ are not allowed?
- Why can clearing denominators introduce extraneous solutions?
- For $\frac{2}{x^2-x}=\frac{1}{x-1}$, what is a sensible first step before multiplying by an LCM?
