Under what situation would one or more solutions of an equation with rational expressions be unacceptable? Need examples.
In an equation with rational expressions, there are certain situations where one or more solutions may be considered unacceptable. This typically occurs when the values of the variables lead to division by zero or result in extraneous solutions.
To better understand this, let's consider two common scenarios where unacceptable solutions may arise:
1. Division by Zero: In rational expressions, the denominator cannot equal zero because division by zero is undefined. Therefore, any value that makes the denominator zero must be excluded from the solution set. For example, consider the equation:
(x - 4) / (x^2 - 16) = 1/x
We need to find the value(s) of x that satisfy the equation. To start, we eliminate the denominator by cross-multiplying:
x - 4 = (x^2 - 16) / x
Next, we multiply both sides of the equation by x to eliminate the fraction:
x(x - 4) = x^2 - 16
Expanding and rearranging, we get:
x^2 - 4x = x^2 - 16
Simplifying further, we have:
-4x = -16
Dividing by -4, we find:
x = 4
However, upon inspection, we notice that x = 4 would make the denominator of the original equation zero, resulting in division by zero. Hence, the solution x = 4 is unacceptable.
2. Extraneous Solutions: Sometimes, when solving equations involving rational expressions, extraneous solutions can appear. These are solutions that, when substituted back into the original equation, do not satisfy the equation.
For example, consider the equation:
(x + 2) / (x - 1) = 1
To solve for x, we cross-multiply:
x + 2 = x - 1
Subtracting x from both sides, we obtain:
2 = -1
This equation has no solution for x. However, when we check the solutions against the original equation, we find that x = 2 makes the denominator zero, which is unacceptable. Hence, x = 2 is an extraneous solution.
It is crucial to be vigilant when solving equations with rational expressions, as division by zero or extraneous solutions can lead to inaccuracies and mistakes. Always double-check the solutions obtained by substituting them into the original equation to ensure their validity.