Questions: The following rational equation has denominators that contain variables. For this equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 4/(x+5) + 2/(x-2) = 14/((x+5)(x-2))

Solution Steps

To solve a rational equation, first identify the values that make any denominator zero to determine the restrictions on the variable. Then, multiply each term by the common denominator to eliminate the fractions and solve the resulting equation, keeping in mind the previously identified restrictions.

To determine the restrictions on the variable \( x \), we need to find the values that make any denominator zero. The denominators are \( x + 5 \) and \( x - 2 \).

\[ x + 5 = 0 \implies x = -5 \] \[ x - 2 = 0 \implies x = 2 \]

Thus, the restrictions on the variable are \( x \neq -5 \) and \( x \neq 2 \).

Next, we multiply each term by the common denominator \((x + 5)(x - 2)\) to eliminate the fractions:

\[ \frac{4}{x+5} \cdot (x+5)(x-2) + \frac{2}{x-2} \cdot (x+5)(x-2) = \frac{14}{(x+5)(x-2)} \cdot (x+5)(x-2) \]

This simplifies to:

\[ 4(x-2) + 2(x+5) = 14 \]

Simplify the equation:

\[ 4x - 8 + 2x + 10 = 14 \]

Combine like terms:

\[ 6x + 2 = 14 \]

Solve the simplified equation for \( x \):

\[ 6x + 2 = 14 \] \[ 6x = 12 \] \[ x = 2 \]

We need to check if the solution \( x = 2 \) is within the restrictions. Since \( x = 2 \) is one of the restricted values, it is not a valid solution.

Final Answer