To solve the inequality [tex]\(\frac{x}{4} \leq \frac{y}{x}\)[/tex], let's go through a detailed, step-by-step solution:
1. Rewrite the inequality: Start with the given inequality:
[tex]\[
\frac{x}{4} \leq \frac{y}{x}
\][/tex]
2. Multiply both sides by [tex]\(x\)[/tex] (assuming [tex]\(x \neq 0\)[/tex]): Multiplying each side by [tex]\(x\)[/tex] eliminates the denominator on the right side:
[tex]\[
\frac{x \cdot x}{4} \leq y
\][/tex]
which simplifies to:
[tex]\[
\frac{x^2}{4} \leq y
\][/tex]
3. Isolate [tex]\(y\)[/tex]: To clearly express the solution set, rewrite the inequality to isolate [tex]\(y\)[/tex]:
[tex]\[
y \geq \frac{x^2}{4}
\][/tex]
Thus, the solution set for the inequality [tex]\(\frac{x}{4} \leq \frac{y}{x}\)[/tex] is given by:
[tex]\[
y \geq \frac{x^2}{4}
\][/tex]
This means that for any value of [tex]\(x\)[/tex], the value of [tex]\(y\)[/tex] must be at least [tex]\(\frac{x^2}{4}\)[/tex] in order to satisfy the inequality.
