Questions: The following rational equation has denominators that contain 1/(x-2) - 3/(x+5) = 7/(x^2 + 3x -10)

Transcript text: The following rational equation has denominators that contain \[ \frac{1}{x-2}-\frac{3}{x+5}=\frac{7}{x^{2}+3 x-10} \]

Solution

Solution Steps

To solve the given rational equation, we need to find a common denominator for all the fractions involved. The denominators are \(x-2\), \(x+5\), and \(x^2 + 3x - 10\). We can factorize \(x^2 + 3x - 10\) to find the common denominator. Once we have the common denominator, we can rewrite each term with this common denominator and then solve the resulting equation.

Step 1: Identify the Common Denominator

The given equation is: \[ \frac{1}{x-2} - \frac{3}{x+5} = \frac{7}{x^2 + 3x - 10} \]

First, factorize the quadratic expression in the denominator on the right-hand side: \[ x^2 + 3x - 10 = (x-2)(x+5) \]

Step 2: Rewrite Each Term with the Common Denominator

The common denominator for all terms is \((x-2)(x+5)\). Rewrite each fraction with this common denominator: \[ \frac{1}{x-2} = \frac{x+5}{(x-2)(x+5)} \] \[ \frac{3}{x+5} = \frac{3(x-2)}{(x-2)(x+5)} \] \[ \frac{7}{x^2 + 3x - 10} = \frac{7}{(x-2)(x+5)} \]

Step 3: Combine the Fractions

Combine the fractions on the left-hand side: \[ \frac{x+5}{(x-2)(x+5)} - \frac{3(x-2)}{(x-2)(x+5)} = \frac{7}{(x-2)(x+5)} \]

Step 4: Simplify the Numerator

Simplify the numerator on the left-hand side: \[ \frac{x+5 - 3(x-2)}{(x-2)(x+5)} = \frac{x+5 - 3x + 6}{(x-2)(x+5)} = \frac{-2x + 11}{(x-2)(x+5)} \]

Step 5: Set the Numerators Equal

Since the denominators are the same, set the numerators equal to each other: \[ -2x + 11 = 7 \]

Step 6: Solve for \( x \)

Solve the equation for \( x \): \[ -2x + 11 = 7 \] \[ -2x = 7 - 11 \] \[ -2x = -4 \] \[ x = 2 \]

Step 7: Check for Extraneous Solutions

Check if \( x = 2 \) is a valid solution by substituting it back into the original equation. Since \( x = 2 \) makes the denominators zero, it is not a valid solution.