This in Precalc we learned how to solve rational equation I choose this subject because it was an interesting application of simplyfiying rational expressions using addition , subtractions, multiplication and division. Solving rational equations is important because they are a useful tools for representing real-life situations and for finding answers to real problems and finding unkowns.
Steps to Solve Rational Equations (note: this is one method to solve Rational Equations using the LCD )
1.Identify the Denominators: Identify all the denominators in the equation if needed find the value for x on the deminator that would make the denomiator equal 0 that is your non-permissable value.
2. Find the Least Common Denominator (LCD): Determine the least common denominator of all the rational expressions in the equation.
3. Clear the Denominators: Multiply both sides of the equation by the LCD to eliminate the denominators. This step turns the rational equation into a polynomial equation.
4. Simplify the Equation: Distribute and combine like terms to simplify the equation.
5. Solve the Resulting Equation: Solve the simplified polynomial equation using appropriate methods (factoring, quadratic formula, etc.).
6. Check for Extraneous Solutions: Substitute the solutions back into the original rational equation to ensure they do not make any denominator zero. Discard any solutions that do.
Example:
Solve the rational equation: .
Step 1: Identify the Denominators
The denominators are and so are non permissible values for x are -1, 2.
Step 2: Find the Least Common Denominator (LCD)
The LCD of and is .
Step 3: Clear the Denominators
Multiply both sides of the equation by the LCD :
Distribute the LCD to each term:
Step 4: Simplify the Equation
Distribute and combine like terms:
Combine like terms on the left side:
Move all terms to one side to set the equation to zero:
Step 5: Solve the Resulting Equation
This is a quadratic equation, so we can solve it using the quadratic formula \, where , , and :
The solutions are and .
Step 6: Check for Extraneous Solutions
Substitute and back into the original equation to ensure they do not make any denominator zero. Since neither value makes or equal to zero, both are valid solutions.
Final Answer
The solutions to the rational equation are and .