Bravo!

This equation has no solutions. The graph shows that doesnottake values between -2 and 0, inclusive, for any value of x. It is also not clear why you need to solve this equation.

This is not clear at all. Which lower number? Why do you consider x = -2?

The absolute value should end after "+2" and before "<", not after "0.5". Vertical bar "|" is used to write absolute value in LaTeX. To make the bars the same vertical size as the formula between them, use \left| ... \right| instead of just | ... |.

Solving the inequality here is a little tricky because one has to multiply by a negative number (which changes the direction of the inequality) and take square of both sides (which introduces extra roots). Therefore, it is better to look at the graph and determine the equation to solve. The red graph of f(x) must be inside the blue band of width 1 (i.e., 2 * 0.5) around the line y = -2. Intersection of which lines do you need to determine?

In fact, the equation is correct, and the relevant root is approximately -5.10. Therefore, if you are looking for an integer N, it would be -6.