Rational Equations: LCD Solutions

• When rational equations are not proportions, determine the    least common denominator   . Then rewrite the problem using the LCD. 

• Before solving for the variable, determine the restrictions on the    denominator   . 

• If a restriction is also a solution, this value is    extraneous    because the denominator would be undefined. 

• After all parts of the problem are written with the LCD, use the    numerator    to solve the problem for the missing value(s) of the variable. 


Note

Keep in mind that if a degree is two or higher, you can only simplify equations if:


• all terms are on the same side of the equation
• the expression is factored

Example 3

Solve. Check your solutions for extraneous values.

$\frac{n}{n+1}+3=\frac{n^2–2}{n+1}$

Plan
Determine the LCD

Simplify the numerator (distribute, combine like terms)

Solve for the unknown

Check

Note

If you need to review how to find LCD, see Lesson 8.

Rational Equations: LCD (cont.) Solutions

• Write    all real numbers    solutions with the math shorthand, ℝ.
• Some rational equations may have    no solution   , which occurs when:

  • all    variables    have simplified out of the equation, and

  • the remaining    constants    are not equal. (i.e. 7 = 12).


Note

Rational equations require perseverance and accuracy. Keep working!

Example 4

Solve. Check your solutions for extraneous values.

$$\begin{gathered} \frac{x}{x+6}+\frac{6}{x–6}=\frac{x^2+36}{x^2–36} \\ \mathrm{LCD}: \left(x–6\right)\left(x+6\right), x\ne \pm 6 \end{gathered}$$

$\begin{array}{rcl}\frac{x\left(x–6\right)}{\left(x+6\right)\left(x–6\right)}+\frac{6\left(x+6\right)}{\left(x+6\right)\left(x–6\right)}& =& \frac{x^2+36}{\left(x+6\right)\left(x–6\right)}\\ & & \\ x^2–6x+6x+36& =& x^2+36\\ x^2+36& =& x^2+36\\ 0& =& 0\\ \mathrm{ℝ}, x& \ne & \pm 6\end{array}$

Example 5

Solve. Check your solutions for extraneous values.