Algebra and Trigonometry, 5th Edition

3.4 Solving Rational Equations and Radical Equations

• Solve rational equations.

• Solve radical equations.

Rational Equations

Equations containing rational expressions are called rational equations. Solving such equations involves multiplying on both sides by the least common denominator (LCD) of all the rational expressions to clear the equation of fractions.

Example 1

Solve: $\frac{x-8}{3}+\frac{x-3}{2}=0$.

Algebraic Solution

We have

$$\begin{array}{rcll}\frac{x-8}{3}+\frac{x-3}{2}& =& 0& \begin{array}{l}\mathbf{\text{The LCD is }}\\ \mathbf{3}\mathbf{\cdot }\mathbf{2}\mathbf{\text{,}}\mathbf{\text{ or }}\mathbf{6.}\end{array}\\ 6(\frac{x-8}{3}+\frac{x-3}{2})& =& 6\cdot 0& \begin{array}{l}\mathbf{\text{Multiplying by }}\\ \mathbf{\text{the LCD on both }}\\ \mathbf{\text{sides to clear fractions}}\end{array}\\ 6\left(\frac{x-8}{3}\right)+6\left(\frac{x-3}{2}\right)& =& 0\\ 2(x-8)+3(x-3)& =& 0\\ 2x-16+3x-9& =& 0\\ 5x-25& =& 0\\ 5x& =& 25\\ x& =& 5.\end{array}$$

The possible solution is 5.

Check:

The solution is 5.

Visualizing ...