Question

Y+3/2x^2-2x<-1/2x^2+4

Answer 1

The answer in terms of y and x is
`\(y < -(2x^2 - x + 6)/(x^2) - 3\)`.

Step 1: Find a common denominator

The common denominator is
`\((2x^2 - 2x)(2x^2 + 4)\)`:

`\[((y + 3)(2x^2 + 4))/((2x^2 - 2x)(2x^2 + 4)) < (-1(2x^2 - 2x))/((2x^2 - 2x)(2x^2 + 4))\]`

Step 2: Distribute and simplify

Expand both the numerator and denominator:

`\[(y + 3)(2x^2 + 4) < -1(2x^2 - 2x)\]`

Step 3: Isolate y

Distribute and combine like terms:

`\[2x^2y + 4y + 6x^2 + 12 < -2x^2 + 2x\]`

Subtract
`\(-2x^2 + 2x\)` from both sides:

`\[2x^2y + 4y + 6x^2 + 12 + 2x^2 - 2x < 0\]`

Combine like terms:

`\[2x^2y + 6x^2 + 4y - 2x + 12 < 0\]`

Step 4: Express the solution for y in terms of x

Factor out
`\(2x^2\)` from the terms with
`\(x^2\)` and factor out 2 from the terms with y:

`\[2x^2(y + 3) + 2(2x^2 - x + 6) < 0\]`

Divide both sides by 2 to simplify:

`\[x^2(y + 3) + 2x^2 - x + 6 < 0\]`

Now, you have the solution expressed for y in terms of x:

`\[y + 3 < -(2x^2 - x + 6)/(x^2)\]`

So, the answer in terms of y and x is
`\(y < -(2x^2 - x + 6)/(x^2) - 3\)`.



The probable question can be:

Question:

Solve the inequality
`\((y + 3)/(2x^2 - 2x) < (-1)/(2x^2 + 4)\)`. Which of the following expressions represents the solution for \(y\) in terms of \(x\)?

a.
`\(y > -(2x^2 - 2x + 4)/(2x^2 + 4)\)`

b.
`\(y < (2x^2 - 2x - 4)/(2x^2 + 4)\)`

c.
`\(y > (2x^2 + 2x - 4)/(2x^2 + 4)\)`

d.
`\(y < -(2x^2 + 2x + 4)/(2x^2 + 4)\)`