Solve the inequality. (Enter your answer using interval notation.) $$ (x-4)(x+5)(3 x+5)

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Solve the inequality. (Enter your answer using interval notation.) $$ (x-4)(x+5)(3 x+5)<0 $$ $$ \square $$

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Solve the equation by following steps: - step0: Solve for $$x$$: $$\left(x-4\right)\left(x+5\right)\left(3x+5\right)=0$$ - step1: Separate into possible cases: $$\begin{align}&x-4=0\&x+5=0\&3x+5=0\end{align}$$ - step2: Solve the equation: $$\begin{align}&x=4\&x=-5\&x=-\frac{5}{3}\end{align}$$ - step3: Rewrite: $$x_{1}=-5,x_{2}=-\frac{5}{3},x_{3}=4$$ To solve the inequality $$(x-4)(x+5)(3x+5)<0$$, we first need to find the critical points where the expression equals zero. Setting the expression equal to zero: $$ (x-4)(x+5)(3x+5) = 0 $$ The solutions to this equation are: 1. $$x - 4 = 0 \Rightarrow x = 4$$ 2. $$x + 5 = 0 \Rightarrow x = -5$$ 3. $$3x + 5 = 0 \Rightarrow x = -\frac{5}{3}$$ Thus, the critical points are $$x = -5$$, $$x = -\frac{5}{3}$$, and $$x = 4$$. Next, we will test the intervals defined by these critical points to determine where the product $$(x-4)(x+5)(3x+5)$$ is negative. The intervals to test are: 1. $$(- \infty, -5)$$ 2. $$(-5, -\frac{5}{3})$$ 3. $$(- \frac{5}{3}, 4)$$ 4. $$(4, \infty)$$ ### Testing the intervals: 1. **Interval $$(- \infty, -5)$$**: Choose $$x = -6$$ $$ (-6-4)(-6+5)(3(-6)+5) = (-10)(-1)(-18) < 0 \quad \text{(negative)} $$ 2. **Interval $$(-5, -\frac{5}{3})$$**: Choose $$x = -4$$ $$ (-4-4)(-4+5)(3(-4)+5) = (-8)(1)(-7) > 0 \quad \text{(positive)} $$ 3. **Interval $$(- \frac{5}{3}, 4)$$**: Choose $$x = 0$$ $$ (0-4)(0+5)(3(0)+5) = (-4)(5)(5) < 0 \quad \text{(negative)} $$ 4. **Interval $$(4, \infty)$$**: Choose $$x = 5$$ $$ (5-4)(5+5)(3(5)+5) = (1)(10)(20) > 0 \quad \text{(positive)} $$ ### Summary of the signs: - $$(- \infty, -5)$$: negative - $$(-5, -\frac{5}{3})$$: positive - $$(- \frac{5}{3}, 4)$$: negative - $$(4, \infty)$$: positive ### Conclusion: The inequality $$(x-4)(x+5)(3x+5)<0$$ holds in the intervals where the product is negative: 1. $$(- \infty, -5)$$ 2. $$(- \frac{5}{3}, 4)$$ Thus, the solution in interval notation is: $$ (-\infty, -5) \cup \left(-\frac{5}{3}, 4\right) $$ The solution to the inequality $$(x-4)(x+5)(3x+5)<0$$ is $$x \in (-\infty, -5) \cup \left(-\frac{5}{3}, 4\right)$$.

Solve the inequality. (Enter your answer using interval notation.)

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Testing the intervals:

Summary of the signs:

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