Questions: Solve for all values of x: 1/(x + 9) + 1/(x - 9) = (x + 2)/(x - 9)

Transcript text: Solve for all values of x: 1/(x + 9) + 1/(x - 9) = (x + 2)/(x - 9)

Solution

Step 1: Combine the fractions on the left side

The given equation is: \[ \frac{1}{x+9} + \frac{1}{9} = \frac{x+2}{9} \]

First, find a common denominator for the fractions on the left side. The common denominator is \(9(x+9)\).

\[ \frac{1}{x+9} + \frac{1}{9} = \frac{9 + (x+9)}{9(x+9)} = \frac{9 + x + 9}{9(x+9)} = \frac{x + 18}{9(x+9)} \]

Step 2: Simplify the equation

Now, the equation becomes: \[ \frac{x + 18}{9(x+9)} = \frac{x+2}{9} \]

Step 3: Cross-multiply to solve for \(x\)

Cross-multiply to eliminate the fractions: \[ (x + 18) \cdot 9 = (x + 2) \cdot 9(x + 9) \]

Simplify both sides: \[ 9(x + 18) = 9(x + 2)(x + 9) \]

Divide both sides by 9: \[ x + 18 = (x + 2)(x + 9) \]

Step 4: Expand and simplify the quadratic equation

Expand the right side: \[ x + 18 = x^2 + 9x + 2x + 18 \] \[ x + 18 = x^2 + 11x + 18 \]

Subtract \(x + 18\) from both sides: \[ 0 = x^2 + 11x + 18 - x - 18 \] \[ 0 = x^2 + 10x \]

Step 5: Factor the quadratic equation

Factor out \(x\): \[ 0 = x(x + 10) \]

Step 6: Solve for \(x\)

Set each factor equal to zero: \[ x = 0 \] \[ x + 10 = 0 \] \[ x = -10 \]

The solutions for \(x\) are: \[ x = 0 \] \[ x = -10 \]

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