Which of the following is the solution set for `(3x-6)(2+x)lt 0` ?

Options

A. `{x : x lt 2}`
B. `{x : x gt 2}`
C. `{x : x gt -2}`
D. `{x : -2 lt x lt 2}`

Answer

D

Explanation

If the product of `(3x-6)` and `(2+x)` is negative, then one of the two factors is negative and the other is positive. There are two cases :
Case 1, when `3x-6 lt 0` and `2+x gt 0` ,
`3x-6 lt 0`
`3x lt 0`
`x lt 2`
and :
`2+x gt 0`
`x gt -2`
So, all x such that `-2 lt x lt 2` satisfy the inequality.
Case 2, when `3x-6 gt0` and `2+x lt 0` :
`3x-6 gt 0`
`3x gt 6`
`x gt 2`
and :
`2+x lt 0`
`x lt -2`
There are no numbers that are both less than - 2 and greater than 2, so the numbers that work in Case 1 are the complete solution set.