Question

How do you solve and write the following in interval notation: `2x^2 - 7x - 4<=0`?

Answer 1

The solution to this inequality is: `[-1/2,4]` (see explanation for an in-depth process of how to solve the inequality).

Explanation:

We start by assuming `<=` is an `=` and solving as such:
`2x^2-7x-4=0`
`(2x+1)(x-4)=0`
`x=-1/2, 4`

These points, `x=-1/2` and `x=4`, the solutions to the equation, give us the bounds of the intervals for the inequality.

When `x<-1/2`, the function `(f(x)=2x^2-7x-4)` is positive.
When `-1/2<` `x<4`, the function is negative.
When `x>4`, the function is positive.

Since we're looking for where `f(x)<=0`, the interval of the solution is `-1/2<=` `x<=4`, which in interval notation is: `[-1/2,4]`.