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Solve the inequality. |x² + 4x + 2| < 2

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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

User Holland Risley
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