Solving Rational Equations Notes

\begin{definition} An \textbf{equation} that contains at least one \textbf{rational expression}. \end{definition}

\begin{note} Rational equations often require finding a \textbf{common denominator} for all the terms involved. The process for solving a rational equation often produces \textbf{extra solutions} that need to be rejected. \end{note}

Solving Rational Equations with a Common Denominator

The simplest rational equations are ones that already have a \textbf{common denominator}.

Solving Rational Equations with a Common Denominator

1. \textbf{Ensure} that all rational expressions have the \textbf{same denominator}.
2. \textbf{Ignore} the denominators and \textbf{equate} the numerators.
3. \textbf{Solve} the resulting equation.
4. \textbf{Check} the solution in the original equation to ensure it doesn't make any denominator zero.

\begin{example} Solve the equation: $$\frac{2x+5}{(x+2)(x-3)} = \frac{13}{(x+2)(x-3)}$$

2. The denominators are already the same.
4. Equate the numerators: $$2x + 5 = 13$$
6. Solve for \$x\$: $$2x = 8 \implies x = 4$$
8. Check the solution: $$\frac{2(4)+5}{(4+2)(4-3)} = \frac{13}{6}$$

The solution is valid. \end{example}

\begin{warning} Always check your solutions to ensure they don't make any denominator zero. Such solutions are called \textbf{extraneous solutions} and must be rejected. \end{warning}

Solving Rational Equations with Cross Multiplication

When a rational equation has a single term on each side with \textbf{different denominators}, \textbf{cross multiplication} can be used.

Solving Rational Equations with Cross Multiplication

1. \textbf{Cross multiply} the numerators and denominators.
2. \textbf{Solve} the resulting equation.
3. \textbf{Check} the solution in the original equation.

\begin{example} Solve the equation: $$\frac{2}{x+1} = \frac{4}{x+4}$$

2. Cross multiply: $$2(x+4) = 4(x+1)$$
4. Solve for \$x\$: $$2x + 8 = 4x + 4 \implies 2x = 4 \implies x = 2$$
6. Check the solution: $$\frac{2}{2+1} = \frac{4}{2+4} \implies \frac{2}{3} = \frac{4}{6}$$

The solution is valid. \end{example}

\begin{note} Cross multiplication is a shortcut for making both sides of the equation have a \textbf{common denominator}. \end{note}

Solving Rational Equations with Multiple Terms

When a rational equation has \textbf{multiple terms} on either side, the terms need to be \textbf{combined} first.

Solving Rational Equations with Multiple Terms

1. \textbf{Find a common denominator} for all terms.
2. \textbf{Combine} the terms on each side.
3. \textbf{Ignore} the denominators and \textbf{equate} the numerators.
4. \textbf{Solve} the resulting equation.
5. \textbf{Check} the solution in the original equation.

\begin{example} Solve the equation: $$\frac{x-1}{x+2} + \frac{x}{x+1} = \frac{6x+5}{x^2+3x+2}$$

2. Find a common denominator: $$(x+2)(x+1)$$
4. Combine the terms: $$\frac{(x+1)(x-1) + (x+2)x}{(x+2)(x+1)} = \frac{6x+5}{(x+2)(x+1)}$$
6. Equate the numerators: $$x^2 - 1 + x^2 + 2x = 6x + 5$$
8. Solve for \$x\$: $$2x^2 + 2x - 1 = 6x + 5 \implies 2x^2 - 4x - 6 = 0$$
10. Factor the quadratic: $$2(x - 3)(x + 1) = 0$$
12. Solve for \$x\$: $$x = 3 \quad \text{or} \quad x = -1$$
14. Check the solutions:

• \$x = 3\$ is valid.
• \$x = -1\$ makes the denominator zero, so it is rejected. \end{example}

\begin{warning} The process of ignoring the denominator can result in \textbf{extraneous solutions}. Always check your solutions in the original equation. \end{warning}

Applications of Rational Equations

Rational equations can be used to solve real-world problems involving \textbf{rates}, \textbf{distances}, and \textbf{proportions}.

\begin{example} Arianna can complete a coding project in 3 hours. Elijah can complete the same project in 2 hours. How long will it take them if they work together?

1. Arianna's rate is \$\frac{1}{3}\$ project per hour.
2. Elijah's rate is \$\frac{1}{2}\$ project per hour.
3. Together, their rate is \$\frac{1}{x}\$ project per hour.

Write the equation: $$\frac{1}{3} + \frac{1}{2} = \frac{1}{x}$$

2. Find a common denominator: $$\frac{2x}{6x} + \frac{3x}{6x} = \frac{6}{6x}$$
4. Solve for \$x\$: $$5x = 6 \implies x = \frac{6}{5} = 1.2$$

It will take them 1.2 hours to complete the project together. \end{example}

\begin{self_review} Solve the following rational equation: $$\frac{x+2}{x-3} + \frac{3}{x+1} = \frac{5x+7}{x^2-2x-3}$$

1. Find a common denominator.
2. Combine the terms.
3. Solve for \$x\$.
4. Check your solution. \end{self_review}

\begin{tok} How do we ensure that the solutions we find for rational equations are valid? What role does \textbf{verification} play in mathematics? \end{tok}