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. $\{x \mid X \ \textless \ -4 \text{ Or } X \ \textgreater \ 2\}$

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Introduction

In mathematics, inequalities are a crucial part of solving equations and understanding the behavior of functions. The given inequality, $5(x-2)(x+4) > 0$, is a quadratic inequality that involves a product of two binomials. To find the solution set, we need to determine the values of $x$ that satisfy the inequality. In this article, we will explore the solution set to the given inequality and compare it with the provided options.

Understanding the Inequality

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

Finding Critical Points

To find the critical points, we need to set the expression $5(x-2)(x+4)$ equal to zero and solve for $x$. We can do this by setting each factor equal to zero and solving for $x$.

$$5(x-2)(x+4) = 0$$

$$x-2 = 0 \quad \text{or} \quad x+4 = 0$$

$$x = 2 \quad \text{or} \quad x = -4$$

These are the critical points, and they divide the number line into three intervals: $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$.

Testing Intervals

To determine which intervals satisfy the inequality, we need to test each interval by substituting a test value into the expression $5(x-2)(x+4)$.

Interval $(-\infty, -4)$

Let's choose a test value, $x = -5$, and substitute it into the expression.

$$5(-5-2)(-5+4) = 5(-7)(-1) = 35 > 0$$

Since the expression is positive, the interval $(-\infty, -4)$ satisfies the inequality.

Interval $(-4, 2)$

Let's choose a test value, $x = 0$, and substitute it into the expression.

$$5(0-2)(0+4) = 5(-2)(4) = -40 < 0$$

Since the expression is negative, the interval $(-4, 2)$ does not satisfy the inequality.

Interval $(2, \infty)$

Let's choose a test value, $x = 3$, and substitute it into the expression.

$$5(3-2)(3+4) = 5(1)(7) = 35 > 0$$

Since the expression is positive, the interval $(2, \infty)$ satisfies the inequality.

Conclusion

Based on the testing of intervals, we can conclude that the solution set to the inequality $5(x-2)(x+4) > 0$ is the union of the intervals $(-\infty, -4)$ and $(2, \infty)$.

Comparison with Options

Let's compare the solution set with the provided options.

A. $\{x \mid x > -4 \text{ and } x < 2\}$

This option is incorrect because it only includes the interval $(2, \infty)$, but not the interval $(-\infty, -4)$.

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

This option is correct because it includes both the interval $(-\infty, -4)$ and the interval $(2, \infty)$.

Final Answer

The final answer is:

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

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Introduction

In the previous article, we explored the solution set to the inequality $5(x-2)(x+4) > 0$. In this article, we will answer some frequently asked questions (FAQs) about the inequality and provide additional insights.

Q&A

Q: What is the significance of the critical points $x = -4$ and $x = 2$?

A: The critical points $x = -4$ and $x = 2$ are the values of $x$ that make the expression $5(x-2)(x+4)$ equal to zero. These points divide the number line into three intervals: $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$.

Q: Why do we need to test each interval?

A: We need to test each interval to determine which intervals satisfy the inequality. By substituting a test value into the expression, we can determine whether the expression is positive or negative in each interval.

Q: What happens if we choose a test value that is not in the interval?

A: If we choose a test value that is not in the interval, we may get a different result. However, this does not affect the solution set, as the test value is not in the interval.

Q: Can we use other methods to solve the inequality?

A: Yes, we can use other methods to solve the inequality, such as graphing the function or using algebraic manipulations. However, the method of testing intervals is a simple and effective way to solve the inequality.

Q: How do we know that the solution set is the union of the intervals $(-\infty, -4)$ and $(2, \infty)$?

A: We know that the solution set is the union of the intervals $(-\infty, -4)$ and $(2, \infty)$ because we tested each interval and found that the expression is positive in both intervals.

Q: What is the relationship between the solution set and the options?

A: The solution set is the union of the intervals $(-\infty, -4)$ and $(2, \infty)$, which corresponds to option B. Option A is incorrect because it only includes the interval $(2, \infty)$.

Q: Can we use the solution set to solve other inequalities?

A: Yes, we can use the solution set to solve other inequalities. By understanding the behavior of the function, we can determine the solution set to other inequalities.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the inequality $5(x-2)(x+4) > 0$. We provided additional insights and clarified some common misconceptions. By understanding the solution set and the behavior of the function, we can solve other inequalities and gain a deeper understanding of mathematical concepts.

Additional Resources

• For more information about inequalities, see the article "Understanding Inequalities: A Comprehensive Guide".
• For more information about quadratic inequalities, see the article "Solving Quadratic Inequalities: A Step-by-Step Guide".
• For more information about graphing functions, see the article "Graphing Functions: A Beginner's Guide".

Final Answer

The final answer is:

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