The inequality given is:

$x |x + 5| \geq -6$

Step 1: Analyze the inequality

Since the absolute value $|x + 5|$ is always non-negative, the left-hand side $x |x + 5|$ will always be greater than or equal to zero. This means that:

$x |x + 5| \geq 0$

This tells us that the left-hand side of the inequality is always non-negative, and since the right-hand side is $-6$, which is negative, the inequality will always hold true. Thus:

$x |x + 5| \geq -6 \quad \text{for all real values of } x.$

Conclusion:

The solution to the inequality is all real numbers.

Do you want further details or have any questions?

5 Related Questions:

1. How do absolute values affect inequalities?
2. What happens if the right-hand side of the inequality is positive?
3. Can this method be used for other inequalities involving absolute values?
4. What is the importance of checking both sides of an inequality?
5. How do we solve absolute value inequalities with quadratic expressions?

Tip:

When dealing with absolute values, remember that they represent the distance from zero, so they are always non-negative.