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INECUACIONES RESOLVER CON EL METODO DE PUNTOS CRITICOS

The inequality given is:

\[
x^2 - 2 \leq 0
\]

### Step 1: Solve the corresponding equation

First, solve the equation:

\[
x^2 - 2 = 0
\]

Add 2 to both sides:

\[
x^2 = 2
\]

Take the square root of both sides:

\[
x = \pm \sqrt{2}
\]

This gives the critical points \(x_1 = -\sqrt{2}\) and \(x_2 = \sqrt{2}\).

### Step 2: Test the intervals

The critical points divide the real line into three intervals:

1. \( (-\infty, -\sqrt{2}) \)
2. \( (-\sqrt{2}, \sqrt{2}) \)
3. \( (\sqrt{2}, \infty) \)

We need to determine where the expression \(x^2 - 2 \leq 0\) is true.

- **Interval \((- \infty, -\sqrt{2})\)**:  
  Choose \(x = -2\):
  \[
  (-2)^2 - 2 = 4 - 2 = 2 > 0 \quad (\text{Not satisfied})
  \]

- **Interval \((- \sqrt{2}, \sqrt{2})\)**:  
  Choose \(x = 0\):
  \[
  0^2 - 2 = -2 < 0 \quad (\text{Satisfied})
  \]

- **Interval \((\sqrt{2}, \infty)\)**:  
  Choose \(x = 2\):
  \[
  2^2 - 2 = 4 - 2 = 2 > 0 \quad (\text{Not satisfied})
  \]

### Step 3: Consider the endpoints

Since the inequality is non-strict (\(\leq\)), the endpoints \(x = \pm \sqrt{2}\) should be included in the solution.

### Final solution:

The solution is the interval:

\[
x \in [-\sqrt{2}, \sqrt{2}]
\]

### Analysis of the Options:

Given that the options are in terms of exact numbers rather than radicals, none of the provided options exactly match the solution \(x_1 = -\sqrt{2}\) and \(x_2 = \sqrt{2}\). Therefore, the correct choice is:

\[
\boxed{\text{NINGUNA DE LAS ANTERIORES}}
\]

---

Would you like more details, or do you have any questions?

Here are 8 related questions for more practice:

1. What changes if the inequality was strict \(x^2 - 2 < 0\)?
2. How would you solve \(x^2 - 2 \geq 0\)?
3. Can you graph the function \(x^2 - 2\) and show the solution visually?
4. What happens if you have \(x^2 - k \leq 0\) for any constant \(k > 0\)?
5. How does this approach generalize to higher degree polynomials?
6. What if the inequality had an absolute value, such as \(|x^2 - 2| \leq 0\)?
7. Can you find the critical points using a different method, like factoring?
8. How would you handle a similar inequality if it involved a cubic equation?

**Tip:** Always verify whether the boundary points satisfy the inequality, especially when dealing with non-strict inequalities (\(\leq\) or \(\geq\)).

Learn how to solve inequalities using the critical points method with a detailed step-by-step solution. This problem involves identifying critical points, testing intervals, and determining the correct interval solution. Suitable for high school students studying inequalities.

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