Solve The Inequality: $(x^2 - 4x)(x + 4) \ \textless \ 0$

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Introduction

In this article, we will delve into solving the inequality $(x^2 - 4x)(x + 4) < 0$. 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 simplify the expression and identify the intervals where the inequality holds true.

Understanding the Inequality

The given inequality is a product of two expressions: $(x^2 - 4x)$ and $(x + 4)$. To solve the inequality, we need to find the values of $x$ that make the product less than zero. This means that either one of the expressions is negative, or both are negative.

Factoring the Quadratic Expression

The first step in solving the inequality is to factor the quadratic expression $(x^2 - 4x)$. We can factor out the common term $x$ from both terms:

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

Now, we can rewrite the original inequality as:

$$(x(x - 4))(x + 4) < 0$$

Finding the Critical Points

To solve the inequality, we need to find the critical points where the expression changes sign. These points occur when the expression is equal to zero or undefined. In this case, the expression is equal to zero when $x = 0$ or $x = 4$. Additionally, the expression is undefined when $x = -4$.

Creating a Sign Chart

To determine the intervals where the inequality holds true, we can create a sign chart. We will evaluate the expression at a point in each interval to determine the sign of the expression.

Interval	$x(x - 4)$	$x + 4$	$(x(x - 4))(x + 4)$
$(-\infty, -4)$	+	-	+
$(-4, 0)$	+	+	+
$(0, 4)$	-	+	-
$(4, \infty)$	-	+	-

Analyzing the Sign Chart

From the sign chart, we can see that the expression $(x(x - 4))(x + 4)$ is negative in the intervals $(-4, 0)$ and $(4, \infty)$. This means that the inequality $(x^2 - 4x)(x + 4) < 0$ is satisfied when $x$ is in these intervals.

Conclusion

In conclusion, the inequality $(x^2 - 4x)(x + 4) < 0$ is satisfied when $x$ is in the intervals $(-4, 0)$ and $(4, \infty)$. We can write this in interval notation as:

$$(-4, 0) \cup (4, \infty)$$

This is the solution to the inequality.

Additional Tips and Tricks

• When solving inequalities, it's essential to consider the signs of the expressions and the critical points.
• Creating a sign chart can help visualize the intervals where the inequality holds true.
• Be careful when evaluating the expression at the critical points, as the expression may be undefined or equal to zero.

Frequently Asked Questions

• Q: What is the solution to the inequality $(x^2 - 4x)(x + 4) < 0$? A: The solution is $(-4, 0) \cup (4, \infty)$.
• Q: How do I create a sign chart for an inequality? A: To create a sign chart, evaluate the expression at a point in each interval to determine the sign of the expression.
• Q: What are the critical points for the inequality $(x^2 - 4x)(x + 4) < 0$? A: The critical points are $x = 0$, $x = 4$, and $x = -4$.

Final Thoughts

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to consider the signs of the expressions and the critical points, and don't be afraid to create a sign chart to visualize the intervals where the inequality holds true. With these tips and tricks, you'll be well on your way to becoming a master of inequality solving.

Introduction

In our previous article, we solved the inequality $(x^2 - 4x)(x + 4) < 0$ and found the solution to be $(-4, 0) \cup (4, \infty)$. However, we know that there are many more inequalities to solve, and each one requires a unique approach. In this article, we will answer some of the most frequently asked questions about inequality solving.

Q&A

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, such as $2x + 3 < 5$. A quadratic inequality, on the other hand, involves a quadratic expression, such as $x^2 + 4x + 4 > 0$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use the following steps:

1. Add or subtract the same value to both sides of the inequality to isolate the variable.
2. Multiply or divide both sides of the inequality by a positive value to eliminate the coefficient of the variable.
3. Write the solution in interval notation.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Find the critical points by setting the quadratic expression equal to zero and solving for the variable.
3. Create a sign chart to determine the intervals where the inequality holds true.
4. Write the solution in interval notation.

Q: What is a sign chart?

A: A sign chart is a table that shows the sign of the expression in each interval. It is a useful tool for determining the intervals where the inequality holds true.

Q: How do I create a sign chart?

A: To create a sign chart, follow these steps:

1. Identify the critical points by setting the expression equal to zero and solving for the variable.
2. Evaluate the expression at a point in each interval to determine the sign of the expression.
3. Write the sign of the expression in each interval in the sign chart.

Q: What are the critical points for a quadratic inequality?

A: The critical points for a quadratic inequality are the values of the variable that make the quadratic expression equal to zero.

Q: How do I find the critical points for a quadratic inequality?

A: To find the critical points for a quadratic inequality, set the quadratic expression equal to zero and solve for the variable.

Q: What is the solution to the inequality $x^2 + 4x + 4 > 0$?

A: The solution to the inequality $x^2 + 4x + 4 > 0$ is $(-\infty, -2) \cup (-2, \infty)$.

Q: What is the solution to the inequality $x^2 - 4x + 4 < 0$?

A: The solution to the inequality $x^2 - 4x + 4 < 0$ is $(2, \infty)$.

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

A: To determine the sign of the expression in each interval, evaluate the expression at a point in each interval.

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

A: A rational inequality is an inequality that involves a rational expression, such as $\frac{x + 2}{x - 1} < 0$. A quadratic inequality, on the other hand, involves a quadratic expression, such as $x^2 + 4x + 4 > 0$.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

1. Factor the numerator and denominator, if possible.
2. Find the critical points by setting the numerator or denominator equal to zero and solving for the variable.
3. Create a sign chart to determine the intervals where the inequality holds true.
4. Write the solution in interval notation.

Conclusion

In this article, we have answered some of the most frequently asked questions about inequality solving. We have covered topics such as linear and quadratic inequalities, sign charts, critical points, and rational inequalities. We hope that this article has been helpful in answering your questions and providing you with a better understanding of inequality solving.

Additional Tips and Tricks

• When solving inequalities, it's essential to consider the signs of the expressions and the critical points.
• Creating a sign chart can help visualize the intervals where the inequality holds true.
• Be careful when evaluating the expression at the critical points, as the expression may be undefined or equal to zero.

Frequently Asked Questions

• Q: What is the solution to the inequality $x^2 + 4x + 4 > 0$? A: The solution to the inequality $x^2 + 4x + 4 > 0$ is $(-\infty, -2) \cup (-2, \infty)$.
• Q: What is the solution to the inequality $x^2 - 4x + 4 < 0$? A: The solution to the inequality $x^2 - 4x + 4 < 0$ is $(2, \infty)$.
• Q: How do I determine the sign of the expression in each interval? A: To determine the sign of the expression in each interval, evaluate the expression at a point in each interval.

Final Thoughts

Inequality solving can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to consider the signs of the expressions and the critical points, and don't be afraid to create a sign chart to visualize the intervals where the inequality holds true. With these tips and tricks, you'll be well on your way to becoming a master of inequality solving.