Explain how to find restrictions on the variable in a rational equation.
The correct answer and explanation is:
To find restrictions on the variable in a rational equation, first identify any values that would make the denominator equal to zero. A rational equation involves fractions with variables in the denominator, and division by zero is undefined in mathematics. Therefore, any value that makes the denominator zero must be excluded from the solution set.
To determine these restrictions, examine each denominator separately. Set each denominator equal to zero and solve for the variable. These solutions are the values that the variable cannot take. For example, if a rational equation has a denominator like x−2x – 2, then set x−2=0x – 2 = 0. Solving gives x=2x = 2. So, x=2x = 2 is a restriction because it would make the denominator zero and the expression undefined.
Sometimes a rational equation has more than one denominator. It is important to find the restrictions for each one. For instance, if the equation has 1x+2x+3=0\frac{1}{x} + \frac{2}{x + 3} = 0, both denominators, xx and x+3x + 3, must be checked. Setting x=0x = 0 and x+3=0x + 3 = 0 gives x=−3x = -3. So, both x=0x = 0 and x=−3x = -3 are restrictions.
Finding these restrictions is necessary to ensure the final solution does not include any value that would create an undefined term. Even if solving the equation algebraically gives an answer that matches a restriction, that answer must be discarded. This step helps avoid incorrect solutions and confirms the work is mathematically valid.
Always write the restrictions clearly before solving the equation. After solving, compare the solutions with the restrictions to make sure only valid answers remain. This process ensures the final answer is accurate and does not produce division by zero.