How is 1/(x-2) = (2x - 6)/(x + 6) + 1 rewritten as 1/(x - 2) = 3x/(x + 6)? I don't understand.
right side
= (2x-6)/(x+6) + 1
= (2x-6)/(x+6) + (x+6)/(x+6)
now we have a common denominator
= (2x-6 + x+6)/(x-6
= 3x/(x+6)
as you required.
To understand how the equation is rewritten, let's go through the steps:
1. Start with the original equation: 1/(x-2) = (2x - 6)/(x + 6) + 1
2. Firstly, let's simplify the right side of the equation. The denominator of the second term is (x + 6), so to add those fractions together, we need a common denominator. We can achieve this by multiplying the second term's numerator and denominator by (x - 2) so that the denominators match:
(2x - 6)/(x + 6) + 1 * (x - 2)/(x - 2) = (2x - 6)(x - 2)/(x + 6)(x - 2) + (x - 2)/(x - 2)
Simplifying, we get: (2x - 6)(x - 2)/(x + 6)(x - 2) + (x - 2)/(x - 2) = [(2x^2 - 10x + 12)/(x^2 + 4x - 12)] + 1
3. Next, we need to find a common denominator for both fractions on the right side of the equation. The common denominator is (x^2 + 4x - 12). So, we rewrite 1 with this common denominator:
1 = (x^2 + 4x - 12)/(x^2 + 4x - 12)
4. Now we can combine the fractions on the right side of the equation:
(2x^2 - 10x + 12)/(x^2 + 4x - 12) + (x^2 + 4x - 12)/(x^2 + 4x - 12) = [(2x^2 - 10x + 12) + (x^2 + 4x - 12)] / (x^2 + 4x - 12)
Simplifying the numerator, we get: 2x^2 - 10x + 12 + x^2 + 4x - 12 = 3x^2 - 6x
5. At this point, we have the equation: 1/(x - 2) = (3x^2 - 6x)/(x^2 + 4x - 12)
6. Now, let's simplify the right side of the equation. We notice that the numerator and denominator both have a common factor of x, so we can cancel out one of the x terms:
3x^2 - 6x = 3x(x - 2)
7. Finally, we substitute the simplified expression back into the equation:
1/(x - 2) = 3x(x - 2)/(x^2 + 4x - 12)
Simplifying the equation, we arrive at the rewritten form: 1/(x - 2) = 3x/(x + 6)
That's how the equation is rewritten as 1/(x - 2) = 3x/(x + 6).