Question

solve the rational inequality. express your answer in interval notation. \\(\frac{-3}{x + 5} \geq \frac{-3}{x - 4}\\) answer 4 points

Explanation:

Step1: Move all terms to left

Subtract $\frac{-3}{x - 4}$ from both sides: $\frac{-3}{x + 5}-\frac{-3}{x - 4}\geq0$. Simplify: $\frac{-3(x - 4)+3(x + 5)}{(x + 5)(x - 4)}\geq0$.

Step2: Simplify numerator

Expand numerator: $-3x + 12 + 3x + 15 = 27$. So inequality becomes $\frac{27}{(x + 5)(x - 4)}\geq0$.

Step3: Find critical points

Denominator zero at $x=-5$ and $x = 4$. These divide number line into intervals: $(-\infty,-5)$, $(-5,4)$, $(4,\infty)$.

Step4: Test intervals

• For $(-\infty,-5)$: Let $x=-6$, $\frac{27}{(-1)(-10)}=\frac{27}{10}>0$, satisfies.
• For $(-5,4)$: Let $x = 0$, $\frac{27}{(5)(-4)}=-\frac{27}{20}<0$, does not satisfy.
• For $(4,\infty)$: Let $x = 5$, $\frac{27}{(10)(1)}=\frac{27}{10}>0$, satisfies.
• Exclude $x=-5$ (undefined) and $x = 4$ (undefined).

Answer:

$(-\infty, -5) \cup (4, \infty)$