Answer:
x = 3
Step-by-step explanation:
Given rational equation:
[tex]\dfrac{6}{x}-\dfrac{9}{x^2-x}=\dfrac{1}{x-1}[/tex]
To solve the given rational equation for x, begin by finding a common denominator for all the fractions.
Factor the denominator of the second fraction:
[tex]\dfrac{6}{x}-\dfrac{9}{x(x-1)}=\dfrac{1}{x-1}[/tex]
Therefore, the least common denominator (LCD) for all the fractions is x(x - 1).
Rewrite the equation with the LCD:
[tex]\dfrac{6(x-1)}{x(x-1)}-\dfrac{9}{x(x-1)}=\dfrac{x}{x(x-1)}[/tex]
Combine the fractions:
[tex]\dfrac{6(x-1)-9}{x(x-1)}=\dfrac{x}{x(x-1)}\\\\\\\\\dfrac{6x-6-9}{x(x-1)}=\dfrac{x}{x(x-1)}\\\\\\\\\dfrac{6x-15}{x(x-1)}=\dfrac{x}{x(x-1)}[/tex]
Multiply both sides by x(x - 1) to cancel the denominators:
[tex]6x-15=x[/tex]
Solve for x:
[tex]6x - 15 - x = x - x\\\\\\5x-15=0\\\\\\5x-15+15=0+15\\\\\\5x=15\\\\\\\dfrac{5x}{5}=\dfrac{15}{5}\\\\\\x=3[/tex]
Therefore, the value of x is:
[tex]\LARGE\boxed{\boxed{x=3}}[/tex]
