Questions: Subtract. x/(5x-6) - (x+3)/(6x) Simplify your answer as much as possible.

Transcript text: Subtract. \[ \frac{x}{5 x-6}-\frac{x+3}{6 x} \] Simplify your answer as much as possible.

Solution

Solution Steps

To subtract the given fractions, we need to find a common denominator. Once we have the common denominator, we can combine the numerators and simplify the resulting expression.

Solution Approach

Identify the common denominator for the fractions.
Rewrite each fraction with the common denominator.
Subtract the numerators.
Simplify the resulting expression.

Step 1: Identify the Fractions

We start with the two fractions: \[ \frac{x}{5x - 6} \quad \text{and} \quad \frac{x + 3}{6x} \]

Step 2: Find the Common Denominator

The common denominator for these fractions is: \[ 6x(5x - 6) \]

Step 3: Rewrite the Fractions

We rewrite each fraction with the common denominator: \[ \frac{x \cdot 6x}{6x(5x - 6)} - \frac{(x + 3)(5x - 6)}{6x(5x - 6)} \]

Step 4: Subtract the Numerators

Now we can combine the fractions: \[ \frac{6x^2 - (x + 3)(5x - 6)}{6x(5x - 6)} \]

Step 5: Expand and Simplify the Numerator

Expanding the numerator: \[ (x + 3)(5x - 6) = 5x^2 - 6x + 15x - 18 = 5x^2 + 9x - 18 \] Thus, the numerator becomes: \[ 6x^2 - (5x^2 + 9x - 18) = 6x^2 - 5x^2 - 9x + 18 = x^2 - 9x + 18 \]

Step 6: Final Expression

The simplified expression is: \[ \frac{x^2 - 9x + 18}{6x(5x - 6)} \]