Solve. $$ \frac{2}{x^{2}+6}\frac{6}{5 x^{2}+18} $$ The solution is $$ \square $$. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

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Solve. $$ \frac{2}{x^{2}+6}>\frac{6}{5 x^{2}+18} $$ The solution is $$ \square $$. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

UpStudy AI · Accepted Answer

Multiply both sides of the inequality by the positive quantity $$(x^2+6)(5x^2+18)$$ (this is positive for all real $$x$$):

$$
\frac{2}{x^2+6} > \frac{6}{5x^2+18} \quad \Longrightarrow \quad 2(5x^2+18) > 6(x^2+6).
$$

Expand both sides:

$$
10x^2 + 36 > 6x^2 + 36.
$$

Subtract $$36$$ from both sides:

$$
10x^2 > 6x^2.
$$

Subtract $$6x^2$$ from both sides:

$$
4x^2 > 0.
$$

Divide by $$4$$ (which is positive):

$$
x^2 > 0.
$$

The inequality $$x^2 > 0$$ holds for all real numbers except $$x = 0$$. In interval notation, the solution is:

$$
(-\infty, 0) \cup (0, \infty).
$$

Thus, the solution is $$\boxed{(-\infty,0)\cup(0,\infty)}$$.
The solution is all real numbers except $$ x = 0 $$, which can be written as $$ (-\infty, 0) \cup (0, \infty) $$.

Solve.

The solution is .
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

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