The given inequality is:

$-x^2 - 25 \leq 0$

Step-by-step solution:

1. First, simplify the inequality: $-x^2 - 25 \leq 0$ Add $25$ to both sides: $-x^2 \leq 25$

2. Next, divide by $-1$ to solve for $x^2$. Remember, dividing by a negative number reverses the inequality: $x^2 \geq -25$

3. Since $x^2$ is always non-negative (i.e., $x^2 \geq 0$ for all real values of $x$), this inequality is always true, because $x^2 \geq -25$ holds for any real number $x$. There is no restriction on the values of $x$.

Thus, the solution to this inequality is all real numbers.

Graph of the solution:

Since the inequality holds for all real numbers, the graph will be a number line where all values from $-\infty$ to $+\infty$ are included.

• The correct option is C, which shows all real numbers included in the solution.

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Related questions:

1. What would the solution be if the inequality was $x^2 \leq 25$?
2. How does squaring affect the direction of inequalities?
3. Can the inequality $x^2 \geq -25$ ever be false?
4. How does the graph change if the inequality was strict (i.e., $-x^2 - 25 < 0$)?
5. What is the significance of $x^2 \geq 0$ in this context?

Tip:

Always remember that squaring a real number results in a non-negative value, which simplifies solving inequalities involving squares.