Question

I’m not sure how to solve this, really need help, would appreciate if you showed work so i could learn how to solve it too thank you

Answer 1

There are two solutions:

`\boxed{v = (4)/(3) \text { and }v = (3)/(2)}\\\\or\\\\\boxed{v = 1.3333.... \text{ and }v= 1.5}\\`

Explanation:

The given equation is :

`(2)/(v-2)=-6\:+\:\frac {1}{v-1}\\\\`

Find the LCM of the denominators:

LCM =
`\left(v-2\right)\left(v-1\right)`

Multiply by LCM:

`(2)/(v-2)\left(v-2\right)\left(v-1\right)=-6\left(v-2\right)\left(v-1\right)+(1)/(v-1)\left(v-2\right)\left(v-1\right)`

Simplify

`(2)/(v-2)\left(v-2\right)\left(v-1\right):\quad 2\left(v-1\right)`

`(1)/(v-1)\left(v-2\right)\left(v-1\right):\quad v-2`

So the equality becomes

`2\left(v-1\right)=-6\left(v-2\right)\left(v-1\right)+v-2`

Expand the brackets:

`2\left(v-1\right):\quad 2v-2`

`\text {Right side becomes: }\\-6\left(v-2\right)\left(v-1\right)+v-2\\Expand \; \left(v-2\right)\left(v-1\right):\quad v^2-3v+2\\\\\text {Right side becomes}\\-6\left(v^2-3v+2\right) + v -2\\= -6v^2+6\cdot \:3v-6\cdot \:2 + v - 2\\\\\\= -6v^2+18v-12+v-2\\\\\text{Simplify}\\= -6v^2+19v-14\\\\\text{Left side}2(v-1= 2v-2\\\\\text{Finally we get}\\2v - 2 = 6v^2+19v-14\\\\`

Switch sides:

`-6v^2+19v-14=2v-2`

Move 2 from right to left(sign changes)

`-6v^2+19v-14 + 2 =2v\\\rightarrow -6v^2+19v-12=2v\\\\`

Move 2v to left side(sign changes)

`-6v^2+19v-12-2v = 0\\\\\rightarrow -6v^2+17v-12=0\\\\`

This is a quadratic equation which can be solved with a quadratic equation calculator. I used a calculator to solve this:

The solutions are:

`v = (4)/(3) \text { and }v = (3)/(2)\\\\\\\text{In decimal these would be}\\\\v = 1.3333.... \text{ and v}= 1.5\\`