To solve the inequality [tex]\(\frac{x}{4} \leq \frac{9}{x}\)[/tex], let's follow a step-by-step approach:
1. Clear the fraction by multiplying both sides by [tex]\(x\)[/tex] (assuming [tex]\(x \neq 0\)[/tex]) to avoid division by zero:
[tex]\[
\frac{x}{4} \leq \frac{9}{x}
\][/tex]
Multiply both sides by [tex]\(4x\)[/tex]:
[tex]\[
x \cdot x \leq 9 \cdot 4
\][/tex]
[tex]\[
x^2 \leq 36
\][/tex]
2. Solve the resulting inequality [tex]\(x^2 \leq 36\)[/tex]:
To solve [tex]\(x^2 \leq 36\)[/tex], we can take the square root of both sides, remembering to consider both the positive and negative roots:
[tex]\[
|x| \leq 6
\][/tex]
This absolute value inequality translates to:
[tex]\[
-6 \leq x \leq 6
\][/tex]
3. Consider the domain of the original inequality:
We must not forget that we multiplied both sides by [tex]\(x\)[/tex]. This step assumes [tex]\(x \neq 0\)[/tex], but we also need to check the case when [tex]\(x\)[/tex] is zero separately to ensure we are capturing all potential solutions.
When [tex]\(x = 0\)[/tex]:
[tex]\[
\frac{0}{4} \leq \frac{9}{0}
\][/tex]
Since [tex]\(\frac{9}{0}\)[/tex] is undefined, [tex]\(x = 0\)[/tex] cannot be a solution.
Therefore, our final solution set must exclude [tex]\(0\)[/tex].
4. Combine the results:
The solution set from step 2 is [tex]\([-6, 6]\)[/tex]. From step 3, we need to exclude [tex]\(x = 0\)[/tex]. Thus, the solution set is:
[tex]\[
[-6, 6] \setminus \{0\}
\][/tex]
In interval notation, this is:
[tex]\[
[-6, 0) \cup (0, 6]
\][/tex]
Therefore, the solution set to the inequality [tex]\(\frac{x}{4} \leq \frac{9}{x}\)[/tex] is:
[tex]\[
\boxed{[-6, 0) \cup (0, 6]}
\][/tex]
