Solve The Following Inequality:${(x+4)^2(x-6)\ \textgreater \ 0}$Write Your Answer As An Interval Or Union Of Intervals. If There Is No Real Solution, Select "No Solution".

Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $(x+4)^2(x-6) > 0$ and express the solution as a union of intervals.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression. The quadratic expression is $(x+4)^2(x-6)$, and we want to find the values of $x$ that make this expression greater than zero.

Factoring the Quadratic Expression

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

$$(x+4)^2(x-6) = (x+4)(x+4)(x-6)$$

We can simplify this expression by combining the two $(x+4)$ terms:

$$(x+4)^2(x-6) = (x+4)^2(x-6)$$

Finding the Critical Points

Critical points are the values of $x$ that make the quadratic expression equal to zero. In this case, the critical points are the values of $x$ that make the expression $(x+4)^2(x-6)$ equal to zero.

We can find the critical points by setting the expression equal to zero and solving for $x$:

$$(x+4)^2(x-6) = 0$$

This equation is true when either $(x+4)^2 = 0$ or $(x-6) = 0$. Solving for $x$, we get:

$$(x+4)^2 = 0 \Rightarrow x+4 = 0 \Rightarrow x = -4$$

$$(x-6) = 0 \Rightarrow x = 6$$

Therefore, the critical points are $x = -4$ and $x = 6$.

Testing Intervals

To find the solution to the inequality, we need to test the intervals between the critical points. The intervals are $(-\infty, -4)$, $(-4, 6)$, and $(6, \infty)$.

We can test each interval by choosing a value of $x$ within the interval and plugging it into the inequality. If the inequality is true for the chosen value, then the entire interval is part of the solution.

Testing the Interval $(-\infty, -4)$

Let's choose a value of $x$ within the interval $(-\infty, -4)$. For example, let's choose $x = -5$.

Plugging $x = -5$ into the inequality, we get:

$$(x+4)^2(x-6) = (-5+4)^2(-5-6) = (-1)^2(-11) = -11$$

Since $-11 < 0$, the inequality is not true for $x = -5$. Therefore, the interval $(-\infty, -4)$ is not part of the solution.

Testing the Interval $(-4, 6)$

Let's choose a value $x$ within the interval $(-4, 6)$. For example, let's choose $x = 0$.

Plugging $x = 0$ into the inequality, we get:

$$(x+4)^2(x-6) = (0+4)^2(0-6) = (4)^2(-6) = -96$$

Since $-96 < 0$, the inequality is not true for $x = 0$. Therefore, the interval $(-4, 6)$ is not part of the solution.

Testing the Interval $(6, \infty)$

Let's choose a value of $x$ within the interval $(6, \infty)$. For example, let's choose $x = 7$.

Plugging $x = 7$ into the inequality, we get:

$$(x+4)^2(x-6) = (7+4)^2(7-6) = (11)^2(1) = 121$$

Since $121 > 0$, the inequality is true for $x = 7$. Therefore, the interval $(6, \infty)$ is part of the solution.

Conclusion

In conclusion, the solution to the inequality $(x+4)^2(x-6) > 0$ is the union of intervals $(6, \infty)$.

Final Answer

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Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to find the values of the variable that make the inequality true. This involves finding the critical points, testing intervals, and expressing the solution as a union of intervals.

Q: What are critical points?

A: Critical points are the values of the variable that make the inequality equal to zero. These points are used to divide the number line into intervals, which are then tested to determine which ones are part of the solution.

Q: How do I find critical points?

A: To find critical points, you need to set the inequality equal to zero and solve for the variable. This will give you the values of the variable that make the inequality equal to zero.

Q: What are intervals?

A: Intervals are the sections of the number line that are created by the critical points. These intervals are then tested to determine which ones are part of the solution.

Q: How do I test intervals?

A: To test intervals, you need to choose a value of the variable within the interval and plug it into the inequality. If the inequality is true for the chosen value, then the entire interval is part of the solution.

Q: What is the solution to an inequality?

A: The solution to an inequality is the union of intervals that make the inequality true. This is expressed as a range of values that satisfy the inequality.

Q: Can an inequality have no solution?

A: Yes, an inequality can have no solution. This occurs when the inequality is always false, meaning that there are no values of the variable that make the inequality true.

Q: How do I express the solution to an inequality?

A: The solution to an inequality is expressed as a union of intervals. This is written in the form $(a, b)$, where $a$ and $b$ are the critical points that divide the number line into intervals.

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

A: A linear inequality is an inequality that involves a linear expression, while a quadratic inequality is an inequality that involves a quadratic expression. Quadratic inequalities are more complex and require more steps to solve.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, it's always a good idea to check your work by hand to ensure that the solution is correct.

Q: How do I graph an inequality?

A: To graph an inequality, you need to plot the critical points on the number line and shade the intervals that make the inequality true. This will give you a visual representation of the solution.

Conclusion

In conclusion, solving inequalities involves finding points, testing intervals, and expressing the solution as a union of intervals. By following these steps, you can solve any inequality and express the solution in a clear and concise manner.

Final Answer

The final answer is $\boxed{(6, \infty)}$.