To solve the inequality [tex]\( 4 - x^2 \leq 0 \)[/tex], let's follow these steps:
1. Rewrite the inequality as an equation to find the boundary points where the inequality might change:
[tex]\[
4 - x^2 = 0
\][/tex]
2. Solve the equation [tex]\( 4 - x^2 = 0 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
4 - x^2 = 0 \implies x^2 = 4 \implies x = \pm 2
\][/tex]
Thus, the solutions to the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
3. Determine the intervals created by these boundary points:
The solutions [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex] divide the number line into three intervals: [tex]\( (-\infty, -2) \)[/tex], [tex]\( [-2, 2] \)[/tex], and [tex]\( (2, \infty) \)[/tex].
4. Test a point in each interval to see where the inequality [tex]\( 4 - x^2 \leq 0 \)[/tex] holds:
- For the interval [tex]\( (-\infty, -2) \)[/tex], pick a point such as [tex]\( x = -3 \)[/tex]:
[tex]\[
4 - (-3)^2 = 4 - 9 = -5 \quad (\text{which is } \leq 0)
\][/tex]
Thus, this interval satisfies the inequality.
- For the interval [tex]\( [-2, 2] \)[/tex], pick a point such as [tex]\( x = 0 \)[/tex] (or check the boundary points directly):
[tex]\[
4 - 0^2 = 4 \quad (\text{which is } \not\leq 0)
\][/tex]
Additionally, checking boundary points:
[tex]\[
4 - (-2)^2 = 4 - 4 = 0 \quad (\text{which is } \leq 0)
\][/tex]
[tex]\[
4 - 2^2 = 4 - 4 = 0 \quad (\text{which is } \leq 0)
\][/tex]
Thus, the values [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex] satisfy the inequality, but points in between do not.
- For the interval [tex]\( (2, \infty) \)[/tex], pick a point such as [tex]\( x = 3 \)[/tex]:
[tex]\[
4 - 3^2 = 4 - 9 = -5 \quad (\text{which is } \leq 0)
\][/tex]
Thus, this interval satisfies the inequality.
5. Combine the intervals where the inequality holds. Thus:
[tex]\[
x \leq -2 \quad \text{or} \quad x \geq 2
\][/tex]
Therefore, the solution to the inequality [tex]\( 4 - x^2 \leq 0 \)[/tex] is:
[tex]\[
x \in (-\infty, -2] \cup [2, \infty)
\][/tex]
