Solving Rational Equations Math Example 2

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Example 2

hard
Solve xx−2−1x+1=3(x−2)(x+1)\frac{x}{x-2} - \frac{1}{x+1} = \frac{3}{(x-2)(x+1)}.

Solution

  1. 1
    Step 1: LCD = (x−2)(x+1)(x-2)(x+1). Multiply through.
  2. 2
    Step 2: x(x+1)−(x−2)=3x(x+1) - (x-2) = 3.
  3. 3
    Step 3: x2+x−x+2=3x^2 + x - x + 2 = 3, so x2=1x^2 = 1, giving x=±1x = \pm 1.
  4. 4
    Step 4: Check x=−1x = -1: denominator (x+1)=0(x+1) = 0. Extraneous! Check x=1x = 1: 1−1−12=−32\frac{1}{-1} - \frac{1}{2} = -\frac{3}{2} and 3(−1)(2)=−32\frac{3}{(-1)(2)} = -\frac{3}{2} ✓

Answer

x=1x = 1
After clearing denominators and solving, always check that solutions don't make any original denominator zero. Here x=−1x = -1 is extraneous because it zeros out (x+1)(x+1).

About Solving Rational Equations

Solving equations that contain rational expressions by multiplying every term by the LCD to clear all denominators, solving the resulting polynomial equation, and checking for extraneous solutions.

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