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Solve simple rational equations

Simple rational equations are equations containing fractions or rational expressions, often proportions or expressions with linear numerators and denominators, that can be solved by identifying values excluded by zero denominators, multiplying by a least common denominator, and solving the resulting equation. The reasoning preserves equivalence only on the valid domain, so proposed solutions must be checked against restrictions; more advanced cases involving high-degree equations, systems, complex rational functions, or asymptotic analysis are not included.

Detailed Explanation: Solve simple rational equations

To solve a simple rational equation:

  1. Identify values that make any denominator zero. These values are excluded.
  2. Find the least common denominator (LCD).
  3. Multiply every term by the LCD to clear the fractions.
  4. Solve the resulting equation.
  5. Check that the answer is not an excluded value.

Example:

2x−1=3x+2\frac{2}{x-1}=\frac{3}{x+2}

First, find the restrictions. The denominators cannot equal zero:

x−1≠0⇒x≠1x-1\ne 0 \quad\Rightarrow\quad x\ne 1 x+2≠0⇒x≠−2x+2\ne 0 \quad\Rightarrow\quad x\ne -2

The LCD is (x−1)(x+2)(x-1)(x+2). Multiply both sides by the LCD:

(x−1)(x+2)(2x−1)=(x−1)(x+2)(3x+2)(x-1)(x+2)\left(\frac{2}{x-1}\right) = (x-1)(x+2)\left(\frac{3}{x+2}\right)

Cancel the common factors:

2(x+2)=3(x−1)2(x+2)=3(x-1)

Distribute:

2x+4=3x−32x+4=3x-3

Solve for xx:

4=x−34=x-3 x=7x=7

Finally, check the restriction. Since 7≠17\ne 1 and 7≠−27\ne -2, the solution is valid.

x=7\boxed{x=7}

Learn by doing: Solve simple rational equations

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    Linear Equation - One Variable, Four Terms

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