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# Extraneous Solutions

If an equation contains a variable in the denominator of a rational expression, then it is possible that the equation may not have a solution.

 For example, letâ€™s try to solve this equation:  Multiply both sides by (x - 4).  Cancel.  Distribute the 2. Simplify the right side. Add 12 to both sides. x - 12x - 12 x - 12 x = (x - 4) Â· 2 - 2x = 2x - 8 - 2x = -8 = 4 The result is x = 4. However, notice what happens when we check the solution. Substitute 4 for x: Simplify.  The result is division by zero, which is undefined.

This means that x = 4 is not a solution.

Therefore, the equation has no solution.

We call 4 an extraneous solution. It is a number that results from solving an equation, but it is not a solution of the equation.

Since it does not satisfy the original equation, an extraneous solution is not really a solution. An extraneous solution is sometimes called a false solution.

Example

Solve: Solution

Multiply each side by x(x + 3), the LCD of the rational expression. Distribute x(x + 3) to each term on the right side. Cancel common factors. 4x = x + 3 - 12 Subtract x from both sides and simplify the right side. 3x = -9 Divide both sides by 3. x = -3 It appears that x = -3 is a solution. However, if we substitute -3 for x in the original equation the result is division by 0. Division by 0 is undefined, so x = -3 is not a solution.

Therefore, the equation has no solution.

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