Subtract Rational 7E64B0

1. **State the problem:** We need to subtract the two rational expressions: $$\frac{x}{2x - 3} - \frac{x + 6}{3x}$$ 2. **Find a common denominator:** The denominators are $2x - 3$ and $3x$. The least common denominator (LCD) is: $$\text{LCD} = (2x - 3)(3x)$$ 3. **Rewrite each fraction with the LCD:** $$\frac{x}{2x - 3} = \frac{x \cdot 3x}{(2x - 3)(3x)} = \frac{3x^2}{(2x - 3)(3x)}$$ $$\frac{x + 6}{3x} = \frac{(x + 6)(2x - 3)}{(3x)(2x - 3)}$$ 4. **Subtract the numerators over the common denominator:** $$\frac{3x^2 - (x + 6)(2x - 3)}{(2x - 3)(3x)}$$ 5. **Expand the numerator:** First expand $(x + 6)(2x - 3)$: $$x \cdot 2x = 2x^2$$ $$x \cdot (-3) = -3x$$ $$6 \cdot 2x = 12x$$ $$6 \cdot (-3) = -18$$ So, $$(x + 6)(2x - 3) = 2x^2 - 3x + 12x - 18 = 2x^2 + 9x - 18$$ 6. **Substitute back and simplify numerator:** $$3x^2 - (2x^2 + 9x - 18) = 3x^2 - 2x^2 - 9x + 18 = x^2 - 9x + 18$$ 7. **Factor the numerator if possible:** $$x^2 - 9x + 18$$ Factors of 18 that sum to -9 are -6 and -3, so: $$x^2 - 9x + 18 = (x - 6)(x - 3)$$ 8. **Write the final expression:** $$\frac{(x - 6)(x - 3)}{(2x - 3)(3x)}$$ This is the simplified form since numerator and denominator have no common factors. **Final answer:** $$\frac{(x - 6)(x - 3)}{3x(2x - 3)}$$