Option 3 : 6x + 5y - 2 > 0

**Given:**

From the question, we can infer the following:

- y is positive.
- x is negative.
- The expression (4 + 5y)/(x - 1) is definitely negative. (∵ Any expression less than - 6 is bound to be negative)

__CONCEPT__

If both sides of the inequality are divided by a negative number, then the sign of the inequality reverses.

For example: If a and b are two natural numbers such that

⇒ a > b

If we multiply both sides of the above inequality by a negative number, say -1, then

⇒ - a < - b

__CALCULATION:__

The expression (4 + 5y)/(x - 1) is definitely negative. (∵ Any expression less than - 6 is bound to be negative)

⇒ 4 + 5y > 0 and x - 1 < 0

According to the question,

(4 + 5y)/(x - 1) < - 6

Multiplying both sides of the inequality by (x - 1), the sign of the inequality reverses.

(4 + 5y) > - 6(x - 1)

⇒ 4 + 5y > - 6x + 6

⇒ 6x + 5y + 4 - 6 > 0

**∴ 6x + 5y - 2 > 0**

__Mistake Points__

In any inequality problem, it is important to be careful about the **sign** of the **number or the expression **that we are multiplying on both sides of the inequality so that the **sign reversal** can be taken care of in case of a **negative expression.**