Graphing Solution Sets to Quadratic Equations Notes

\begin{definition}[Rational Equation] An equation that contains rational expressions. \end{definition}

\begin{note} Rational equations can have one or more solutions. However, some solutions may be extraneous (not valid) because they make the denominator zero. \end{note}

Solving Rational Equations

To solve rational equations:

1. Eliminate the denominators by multiplying both sides by the least common denominator (LCD).
2. Solve the resulting equation.
3. Check for extraneous solutions by substituting back into the original equation.

\begin{example} Solve the equation:

$$\frac{2}{x} + \frac{3}{x+1} = \frac{5}{x(x+1)}$$

2. The LCD is $x(x+1)$. Multiply both sides by the LCD:
3. $$x(x+1)\left(\frac{2}{x} + \frac{3}{x+1}\right) = x(x+1)\left(\frac{5}{x(x+1)}\right)$$
5. Simplify:
6. $$2(x+1) + 3x = 5$$
8. Solve the equation:
9. $$2x + 2 + 3x = 5$$
10. $$5x + 2 = 5$$
11. $$5x = 3$$
12. $$x = \frac{3}{5}$$
14. Check for extraneous solutions by substituting $x = \frac{3}{5}$ back into the original equation. Since it does not make any denominator zero, it is a valid solution. \end{example}

\begin{warning} Extraneous solutions are solutions that arise during the solving process but do not satisfy the original equation. Always check your solutions! \end{warning}

Applications of Rational Equations

Rational equations are used in various real-world applications, such as:

• Rate problems (e.g., work, speed)
• Proportions and ratios
• Mixture problems

\begin{example} Rate Problem:

A pipe can fill a tank in 3 hours, and another pipe can fill the same tank in 6 hours. How long will it take to fill the tank if both pipes are used together?

2. Let $x$ be the time in hours to fill the tank together.
4. The rate of the first pipe is $\frac{1}{3}$ tanks per hour, and the second pipe is $\frac{1}{6}$ tanks per hour.
6. The equation is:
7. $$\frac{1}{3}x + \frac{1}{6}x = 1$$
9. The LCD is 6. Multiply both sides by 6:
10. $$2x + x = 6$$
12. Solve the equation:
13. $$3x = 6$$
14. $$x = 2$$
16. It takes 2 hours to fill the tank together. \end{example}

\begin{self_review} Solve the equation:

$$\frac{3}{x-2} = \frac{5}{x+1}$$

Check for extraneous solutions. \end{self_review}

\begin{tok} How do we ensure that the solutions we find are valid? What role does checking for extraneous solutions play in the reliability of mathematical methods? \end{tok}