What Is The Solution Set To The Inequality $5(x-2)(x+4)\ \textgreater \ 0$?A. \$\{x \mid X \ \textgreater \ -4$ And $x < 2\}$ B. \$\{x \mid X \ \textless \ -4$ Or $x > 2\}$ 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 break down the problem step by step, using algebraic techniques to identify the intervals where the inequality holds true.

Understanding the Inequality

The given inequality is $5(x-2)(x+4) > 0$. To solve this, we need to find the values of $x$ that make the expression $5(x-2)(x+4)$ positive. We can start by identifying the critical points, which are the values of $x$ that make the expression equal to zero.

Critical Points

The expression $5(x-2)(x+4)$ will be equal to zero when either $(x-2)$ or $(x+4)$ is equal to zero. This gives us two critical points: $x = 2$ and $x = -4$.

Testing Intervals

To determine the solution set, we need to test the intervals between the critical points. We will choose test points within each interval and substitute them into the original inequality to see if it holds true.

Interval 1: $x < -4$

Let's choose a test point $x = -5$. Substituting this into the inequality, we get:

$5(-5-2)(-5+4) > 0$

Simplifying, we get:

$5(-7)(-1) > 0$

This simplifies to:

$35 > 0$

Since this is true, the inequality holds true for all values of $x$ less than $-4$.

Interval 2: $-4 < x < 2$

Let's choose a test point $x = 0$. Substituting this into the inequality, we get:

$5(0-2)(0+4) > 0$

Simplifying, we get:

$5(-2)(4) > 0$

This simplifies to:

$-40 > 0$

Since this is false, the inequality does not hold true for all values of $x$ between $-4$ and $2$.

Interval 3: $x > 2$

Let's choose a test point $x = 3$. Substituting this into the inequality, we get:

$5(3-2)(3+4) > 0$

Simplifying, we get:

$5(1)(7) > 0$

This simplifies to:

$35 > 0$

Since this is true, the inequality holds true for all values of $x$ greater than $2$.

Conclusion

Based on our analysis, we can conclude that the solution set to the inequality $5(x-2)(x+4) > 0$ is the union of two intervals: $x < -4$ and $x > 2$. This can be expressed as:

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

This is option B in the given choices.

Final Answer

The final answer is:

\{x \mid x < -4 \text{ or } x > 2\}$<br/> # Frequently Asked Questions (FAQs) on the Solution Set to the Inequality $5(x-2)(x+4) > 0$ ## Introduction In our previous article, we explored the solution set to the inequality $5(x-2)(x+4) > 0$. We identified the critical points, tested the intervals, and concluded that the solution set is the union of two intervals: $x < -4$ and $x > 2$. In this article, we will address some frequently asked questions (FAQs) related to the solution set. ## Q: What are the critical points of the inequality $5(x-2)(x+4) > 0$? A: The critical points are the values of $x$ that make the expression $5(x-2)(x+4)$ equal to zero. These are $x = 2$ and $x = -4$. ## Q: How do I determine the solution set to the inequality $5(x-2)(x+4) > 0$? A: To determine the solution set, you need to identify the critical points, test the intervals between the critical points, and choose test points within each interval. You then substitute the test points into the original inequality to see if it holds true. ## Q: What is the solution set to the inequality $5(x-2)(x+4) > 0$? A: The solution set is the union of two intervals: $x < -4$ and $x > 2$. This can be expressed as: $\{x \mid x < -4 \text{ or } x > 2\}

Q: Why is the inequality $5(x-2)(x+4) > 0$ true for all values of $x$ less than $-4$?

A: This is because when $x < -4$, both $(x-2)$ and $(x+4)$ are negative. When you multiply two negative numbers, the result is positive. Therefore, the inequality holds true for all values of $x$ less than $-4$.

Q: Why is the inequality $5(x-2)(x+4) > 0$ true for all values of $x$ greater than $2$?

A: This is because when $x > 2$, both $(x-2)$ and $(x+4)$ are positive. When you multiply two positive numbers, the result is positive. Therefore, the inequality holds true for all values of $x$ greater than $2$.

Q: What is the relationship between the solution set and the critical points?

A: The solution set is the union of the intervals between the critical points. In this case, the critical points are $x = 2$ and $x = -4$, and the solution set is the union of the intervals $x < -4$ and $x > 2$.

Q: How do I express the solution set in interval notation?

A: The solution set can be expressed in interval notation as:

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

This is read as "the set of all $x$ such that $x$ is less than $-4$ or $x$ is greater than $2$".

Conclusion

In this article, we addressed some frequently asked questions (FAQs) related to the solution set to the inequality $5(x-2)(x+4) > 0$. We hope that this article has provided you with a better understanding of the solution set and how to determine it.

Final Answer

The final answer is:

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