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

Apr 1, 2017

The solution to this inequality is: $\left[- \frac{1}{2} , 4\right]$ (see explanation for an in-depth process of how to solve the inequality).

#### Explanation:

We start by assuming $\le$ is an $=$ and solving as such:
$2 {x}^{2} - 7 x - 4 = 0$
$\left(2 x + 1\right) \left(x - 4\right) = 0$
$x = - \frac{1}{2} , 4$

These points, $x = - \frac{1}{2}$ and $x = 4$, the solutions to the equation, give us the bounds of the intervals for the inequality.

When $x < - \frac{1}{2}$, the function $\left(f \left(x\right) = 2 {x}^{2} - 7 x - 4\right)$ is positive.
When $- \frac{1}{2} <$$x < 4$, the function is negative.
When $x > 4$, the function is positive.

Since we're looking for where $f \left(x\right) \le 0$, the interval of the solution is $- \frac{1}{2} \le$$x \le 4$, which in interval notation is: $\left[- \frac{1}{2} , 4\right]$.