To solve the inequality $|x| > -1$, we need to consider the properties of absolute values and inequalities.

Step 1: Understand the inequality

The absolute value $|x|$ represents the distance of $x$ from 0 on the number line. By definition:

$|x| \geq 0$

This means that $|x|$ is always non-negative. The smallest value $|x|$ can take is 0 (when $x = 0$).

Step 2: Analyze the inequality

The inequality is $|x| > -1$. Since $|x|$ is always non-negative and $-1$ is negative, $|x|$ will always be greater than $-1$ for any real number $x$.

Conclusion:

Since $|x|$ is always greater than $-1$ for any real number $x$, the solution is:

$\boxed{\text{All real numbers} \ x \in \mathbb{R}}$

This means every real number satisfies the inequality $|x| > -1$.

Would you like details on this explanation?

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Here are five related questions:

1. What if the inequality were $|x| \geq 0$ instead of $|x| > -1$?
2. How would the solution change if the inequality were $|x| > 1$ instead of $|x| > -1$?
3. What are the implications if the inequality was $|x| < 1$ instead?
4. How would the solution change if the inequality were $|x| \geq 2$?
5. What is the effect of changing the sign inside the absolute value, such as $|-x| > -1$?

Tip: Remember that the absolute value of any real number is always non-negative, so inequalities involving absolute values often have different properties from regular linear inequalities.