What Is The Solution Set To The Inequality $5(x-2)(x+4)\ \textgreater \ 0$?A. $\{x \mid X\ \textgreater \ -4 \text{ And } X\ \textless \ 2\}$B. $\{x \mid X\ \textless \ -4 \text{ Or } X\ \textgreater \ 2\}$C. $\{x

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Introduction

In this article, we will explore the solution set to the inequality $5(x-2)(x+4) > 0$. This involves finding the values of $x$ that satisfy the given inequality. We will use algebraic techniques to solve the inequality and determine the solution set.

Understanding the Inequality

The given inequality is $5(x-2)(x+4) > 0$. This is a quadratic inequality, which involves a quadratic expression and a constant. The quadratic expression is $5(x-2)(x+4)$, and the constant is $0$. The inequality states that the quadratic expression is greater than $0$.

Factoring the Quadratic Expression

To solve the inequality, we need to factor the quadratic expression. The quadratic expression can be factored as follows:

$$5(x-2)(x+4) = 5(x^2 + 2x - 8)$$

Finding the Critical Points

The critical points of the inequality are the values of $x$ that make the quadratic expression equal to $0$. To find the critical points, we need to set the quadratic expression equal to $0$ and solve for $x$.

$$5(x^2 + 2x - 8) = 0$$

Dividing both sides by $5$, we get:

$$x^2 + 2x - 8 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = 2$, and $c = -8$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)}$$

Simplifying, we get:

$$x = \frac{-2 \pm \sqrt{36}}{2}$$

$$x = \frac{-2 \pm 6}{2}$$

This gives us two possible values for $x$:

$$x = \frac{-2 + 6}{2} = 2$$

$$x = \frac{-2 - 6}{2} = -4$$

Determining the Solution Set

Now that we have found the critical points, we can determine the solution set to the inequality. The critical points are $x = -4$ and $x = 2$. These points divide the number line into three intervals: $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$.

To determine the solution set, we need to test a value from each interval. Let's choose the following values:

• From the interval $(-\infty, -4)$, let's choose $x = -5$.
• From the interval $(-4, 2)$, let's choose $x = 0$.
• From the interval $(2, \infty)$, let's choose $x = 3$.

Plugging these values into the quadratic expression, we get:

• For $x = -5$, we get: $5((-5)^2 + 2(-5) - 8) = 5(25 - 10 - 8) = 5(7) = 35 > 0$
• For $x = 0$, we get: $5((0)^2 + 2(0) - 8) = 5(-8) = -40 < 0$
• For $x = 3$, we get: $5((3)^2 + 2(3) - 8) = 5(9 + 6 - 8) = 5(7) = 35 > 0$

Since the quadratic expression is positive for $x = -5$ and $x = 3$, and negative for $x = 0$, we can conclude that the solution set to the inequality is:

$$\{x \mid x < -4 \text{ or } x > 2\}$$

Conclusion

In this article, we have explored the solution set to the inequality $5(x-2)(x+4) > 0$. We have used algebraic techniques to solve the inequality and determine the solution set. The solution set is $\{x \mid x < -4 \text{ or } x > 2\}$. This means that the values of $x$ that satisfy the inequality are all real numbers less than $-4$ and all real numbers greater than $2$.

Final Answer

The final answer is $\boxed{B}$

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Introduction

In our previous article, we explored the solution set to the inequality $5(x-2)(x+4) > 0$. We used algebraic techniques to solve the inequality and determine the solution set. In this article, we will answer some frequently asked questions (FAQs) about the solution set.

Q: What is the solution set to the inequality $5(x-2)(x+4) > 0$?

A: The solution set to the inequality $5(x-2)(x+4) > 0$ is $\{x \mid x < -4 \text{ or } x > 2\}$. This means that the values of $x$ that satisfy the inequality are all real numbers less than $-4$ and all real numbers greater than $2$.

Q: Why is the solution set $\{x \mid x < -4 \text{ or } x > 2\}$?

A: The solution set is $\{x \mid x < -4 \text{ or } x > 2\}$ because the quadratic expression $5(x-2)(x+4)$ is positive for all real numbers less than $-4$ and all real numbers greater than $2$. This is because the quadratic expression is a product of two factors, $(x-2)$ and $(x+4)$, and the product of two positive numbers is positive.

Q: What are the critical points of the inequality $5(x-2)(x+4) > 0$?

A: The critical points of the inequality $5(x-2)(x+4) > 0$ are $x = -4$ and $x = 2$. These points divide the number line into three intervals: $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$.

Q: How do I determine the solution set to a quadratic inequality?

A: To determine the solution set to a quadratic inequality, you need to find the critical points of the inequality and test a value from each interval. If the quadratic expression is positive for a value from an interval, then the entire interval is part of the solution set.

Q: Can I use a graphing calculator to solve a quadratic inequality?

A: Yes, you can use a graphing calculator to solve a quadratic inequality. Graphing calculators can help you visualize the solution set and make it easier to determine the critical points.

Q: What is the difference between a quadratic inequality and a quadratic equation?

A: A quadratic equation is an equation that involves a quadratic expression and is equal to $0$. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression and is not equal to $0$. The solution set to a quadratic inequality is a set of values that satisfy the inequality.

Q: Can I solve a quadratic inequality using factoring?

A: Yes, you can solve a quadratic inequality using factoring. Factoring involves expressing the quadratic expression as a product of two or more factors. Once you have factored the quadratic expression, you can use the factors to determine the critical points and test a value from each interval.

Q: What are some common mistakes to avoid when solving a quadratic inequality?

A: Some common mistakes to avoid when solving a quadratic inequality include:

• Not finding the critical points of the inequality
• Not testing a value from each interval
• Not considering the sign of the quadratic expression in each interval
• Not using the correct method to solve the inequality (e.g. factoring, graphing, etc.)

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about the solution set to the inequality $5(x-2)(x+4) > 0$. We have discussed the solution set, critical points, and methods for solving quadratic inequalities. We hope that this article has been helpful in clarifying any questions you may have had about quadratic inequalities.