Equations with Rational Expressions - Step by Step Tutorial

This tutorial explains how to solve equations that contain rational expressions. The main strategy is to multiply all terms of the equation by the lowest common denominator (LCD) to eliminate fractions.

Examples with Solutions

Example 1

Solve the following equation:

\[ 1 - \frac{1}{x - 2} = 4 \]

Solution to Example 1

  1. Multiply all terms by the denominator \(x - 2\) to eliminate the fraction: \[ (x - 2)\left(1 - \frac{1}{x - 2}\right) = (x - 2) \cdot 4 \]
  2. Simplify each side: \[ (x - 2) - 1 = 4(x - 2) \]
  3. Expand and simplify: \[ x - 3 = 4x - 8 \]
  4. Move all terms containing \(x\) to one side: \[ -3x - 3 = -8 \]
  5. Isolate \(x\): \[ -3x = -5 \] \[ x = \frac{5}{3} \]

Conclusion: The solution is \[ x = \frac{5}{3}. \]

Example 2

Find all real solutions to the equation:

\[ 1 - \frac{1}{x - 2} = -\frac{4}{x^2 - 4} \]

Solution to Example 2

  1. Identify the lowest common denominator (LCD): \((x-2)(x+2)\). Multiply all terms by the LCD: \[ (x-2)(x+2)\left(1 - \frac{1}{x - 2}\right) = (x-2)(x+2)\left(-\frac{4}{x^2 - 4}\right) \]
  2. Cancel common factors: \[ (x-2)(x+2) - (x+2) = -4 \]
  3. Expand and simplify: \[ x^2 - x - 6 = -4 \]
  4. Add 4 to both sides: \[ x^2 - x - 2 = 0 \]
  5. Factor the quadratic: \[ (x + 1)(x - 2) = 0 \]
  6. Set each factor to zero and solve: \[ x_1 = -1, \quad x_2 = 2 \]
  7. Check each solution:

    Check \(x = -1\): \[ \text{Left Side} = 1 - \frac{1}{-1 - 2} = 1 - \frac{1}{-3} = \frac{4}{3} \] \[ \text{Right Side} = -\frac{4}{(-1)^2 - 4} = -\frac{4}{1-4} = \frac{4}{3} \] Valid solution.

    Check \(x = 2\): This value makes the denominator \(x-2=0\), which is undefined. Therefore, it is not a valid solution.

Conclusion: The solution is \[ x = -1 \].

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