Final answer:
To solve the inequality |x² + 4x + 2| < 2, we split it into two inequalities and solve each separately. The solution is x < 0 and -4 < x < 0.
Step-by-step explanation:
To solve the inequality |x² + 4x + 2| < 2, we need to break it into two separate inequalities:
x² + 4x + 2 < 2 and -(x² + 4x + 2) < 2
Solving each inequality gives us:
x² + 4x < 0 and -x² - 4x - 4 < 2
For the first inequality, we can factor out x:
x(x + 4) < 0
This means that either x < 0 and x + 4 > 0, or x > 0 and x + 4 < 0. Simplifying these conditions gives us:
-4 < x < 0
For the second inequality, we can simplify it by removing the negative sign:
x² + 4x + 6 < 2
x² + 4x + 4 < 0
This quadratic equation has no real solutions, so the second inequality is not possible.
Therefore, the solution to the original inequality is:
x < 0 and -4 < x < 0