Solve The Following Inequality:$ (x-2)(x+4)(x+7) \geq 0 $Write Your Answer As An Interval Or Union Of Intervals. If There Is No Real Solution, Click On No Solution.

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Introduction

In this article, we will focus on solving the given inequality $(x-2)(x+4)(x+7) \geq 0$. This involves finding the values of $x$ that satisfy the given inequality. We will use various mathematical techniques to solve this inequality and express the solution as a union of intervals.

Understanding the Inequality

The given inequality is a product of three factors: $(x-2)$, $(x+4)$, and $(x+7)$. To solve this inequality, we need to find the values of $x$ that make the product of these factors greater than or equal to zero.

Finding the Critical Points

To solve the inequality, we need to find the critical points where the expression $(x-2)(x+4)(x+7)$ changes sign. These critical points occur when any of the factors $(x-2)$, $(x+4)$, or $(x+7)$ equals zero.

• The first factor $(x-2)$ equals zero when $x = 2$.
• The second factor $(x+4)$ equals zero when $x = -4$.
• The third factor $(x+7)$ equals zero when $x = -7$.

Creating a Sign Chart

To determine the intervals where the inequality is satisfied, we create a sign chart. We examine the sign of the expression $(x-2)(x+4)(x+7)$ in each interval defined by the critical points.

Interval	$(x-2)$	$(x+4)$	$(x+7)$	$(x-2)(x+4)(x+7)$
$(-\infty, -7)$	$-$	$-$	$-$	$-$
$(-7, -4)$	$-$	$-$	$+$	$+$
$(-4, 2)$	$-$	$+$	$+$	$-$
$(2, \infty)$	$+$	$+$	$+$	$+$

Analyzing the Sign Chart

From the sign chart, we can see that the expression $(x-2)(x+4)(x+7)$ is positive in the intervals $(-7, -4)$ and $(2, \infty)$. It is also negative in the intervals $(-\infty, -7)$ and $(-4, 2)$.

Expressing the Solution as Intervals

Based on the sign chart, we can express the solution to the inequality as a union of intervals:

$(-7, -4) \cup (2, \infty)$

This means that the inequality $(x-2)(x+4)(x+7) \geq 0$ is satisfied for all values of $x$ in the intervals $(-7, -4)$ and $(2, \infty)$.

Conclusion

In this article, we solved the inequality $(x-2)(x+4)(x+7) \geq 0$ by finding the critical points and creating a sign chart. We expressed the solution as a union of intervals and determined that the inequality is satisfied for all values of $x$ in the intervals $(-7, -4)$ and $(2, \infty)$.

Frequently Asked Questions

Q: What is the solution to the inequality $(x-2)(x+4)(x+7) \geq 0$?

A: The solution to the inequality is $(-7, -4) \cup (2, \infty)$.

Q: How do I find the critical points of the inequality?

A: To find the critical points, set each factor of the expression equal to zero and solve for $x$.

Q: What is the purpose of creating a sign chart?

A: The sign chart helps us determine the intervals where the inequality is satisfied.

Q: Can the inequality have no real solution?

A: No, the inequality has real solutions in the intervals $(-7, -4)$ and $(2, \infty)$.

Final Answer

The final answer is: $\boxed{(-7, -4) \cup (2, \infty)}$

Introduction

In our previous article, we solved the inequality $(x-2)(x+4)(x+7) \geq 0$ by finding the critical points and creating a sign chart. In this article, we will answer some frequently asked questions related to solving inequalities.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to find the critical points. Critical points occur when the expression on one side of the inequality equals zero.

Q: How do I find the critical points of an inequality?

A: To find the critical points, set each factor of the expression equal to zero and solve for $x$. For example, if the inequality is $(x-2)(x+4)(x+7) \geq 0$, set each factor equal to zero and solve for $x$: $x-2=0$, $x+4=0$, and $x+7=0$.

Q: What is the purpose of creating a sign chart?

A: The sign chart helps us determine the intervals where the inequality is satisfied. By examining the sign of the expression in each interval, we can determine where the inequality is true.

Q: Can the inequality have no real solution?

A: Yes, an inequality can have no real solution. This occurs when the expression on one side of the inequality is always negative, or when the expression is always positive.

Q: How do I determine the sign of an expression in an interval?

A: To determine the sign of an expression in an interval, examine the sign of each factor in the expression. If the factor is positive, the expression is positive. If the factor is negative, the expression is negative.

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

A: A linear inequality is an inequality that can be written in the form $ax+b \geq 0$ or $ax+b \leq 0$, where $a$ and $b$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2+bx+c \geq 0$ or $ax^2+bx+c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, first factor the quadratic expression, if possible. Then, find the critical points by setting each factor equal to zero and solving for $x$. Finally, create a sign chart to determine the intervals where the inequality is satisfied.

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 express the solution to an inequality as intervals?

A: To express the solution to an inequality as intervals, use the following format: $[a, b] \cup [c, d]$, where $a$, $b$, $c$, and $d$ are the critical points.

Conclusion

In this article, we answered some frequently asked questions related to solving inequalities. We covered topics such as finding critical points, creating sign charts, and expressing solutions as intervals. By following these steps, you can solve inequalities and express the solutions as intervals.

Frequently Asked Questions

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

A: A linear inequality is an inequality that can be written in the form $ax+b \geq 0$ or $ax+b \leq 0$, where $a$ and $b$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2+bx+c \geq 0$ or $ax^2+bx+c \leq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, first factor the quadratic expression, if possible. Then, find the critical points by setting each factor equal to zero and solving for $x$. Finally, create a sign chart to determine the intervals where the inequality is satisfied.

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.

Final Answer

The final answer is: $\boxed{(-7, -4) \cup (2, \infty)}$