Solving Rational Equations

5.6 Rational Equations

  • Focus: Solving equations involving rational expressions.

Key Steps to Solve Rational Equations

  1. Find the LCD (Lowest Common Denominator)

    • Identify the LCD of all rational expressions in the equation.

  2. Multiply Both Sides by the LCD

    • This clears the fractions early in the solving process.

  3. Solve the Resulting Equation

    • Simplify and solve the equation after multiplying by the LCD.

  4. Check Solutions

    • Verify none of the solutions make the denominator zero.

    • Identifying any invalid solutions is crucial; these are called extraneous solutions.

    • If all solutions are extraneous, the equation has no valid solution.

Example Walkthrough

Example 1:

  • a.

    1. Find LCD: 15x

    2. Multiply both sides by LCD:

      • Result: Simplified multiplication across the equation.

    3. Reduce & Solve: Resulting equation can be simplified further.

    4. Check: Ensure no denominators are zero with found value.

      • Accepted solution set: {solution found}.

  • b.

    1. Find LCD after factoring: LCD

    2. Multiply and distribute: Clear fractions in both terms.

    3. Solve: Work through simplified equation.

    4. Check for extraneous solutions.

      • Accepted solution set: {solution found}.

  • c.

    1. Factor denominators to find the LCD.

    2. Reduce before multiplying to keep it simple.

    3. Solve resulting equation after distributing.

    4. Check original denominators for zeros.

      • Accepted solution set: {solution found}.

Example 2: More Complex Cases

Example 2:

  • a.

    1. Identify LCD: LCD

    2. Multiply both sides by LCD.

    3. Check solutions for validity:

      • Discovered -4 makes the denominator zero - extraneous solution, discard.

    4. Conclusion: No valid solution exists for this case.

  • b.

    1. Identify simpler LCD: LCD

    2. Solve as in prior examples.

    3. Check both solutions and discard any making denominators zero:

      • Solution: {2} (valid), discard {3} (extraneous).

  • c.

    1. Identify larger LCD: LCD

    2. Simplify and solve.

    3. Check solutions for zero denominators.

      • Valid solution: {6} accepted.

Summary of Exercises

  • Practice your skills: Solve equations numbered 1 through 50 following the methods outlined above.