Let's Solve The Inequality $-x^2 - 5x - 6 \ \textless \ 0$.1. For Which Values Of The Variable $x$ Is The Value Of The Expression $-x^2 - 5x - 6$ Positive?2. For Which Values Of The Variable $x$ Is The Value Of The

Introduction

In this article, we will delve into the world of quadratic inequalities and explore the solution to the given inequality $-x^2 - 5x - 6 < 0$. We will examine the values of the variable $x$ for which the expression $-x^2 - 5x - 6$ is positive. To achieve this, we will employ various mathematical techniques, including factoring, the quadratic formula, and graphing.

Understanding the Quadratic Expression

The given quadratic expression is $-x^2 - 5x - 6$. This expression can be factored as $-(x + 3)(x + 2)$. Factoring the expression allows us to identify its roots, which are the values of $x$ that make the expression equal to zero.

Factoring the Quadratic Expression

To factor the quadratic expression, we need to find two numbers whose product is $-6$ and whose sum is $-5$. These numbers are $-3$ and $-2$, as their product is $-6$ and their sum is $-5$. Therefore, we can write the quadratic expression as $-(x + 3)(x + 2)$.

Identifying the Roots of the Quadratic Expression

The roots of the quadratic expression are the values of $x$ that make the expression equal to zero. In this case, the roots are $x = -3$ and $x = -2$. These values of $x$ make the expression $-(x + 3)(x + 2)$ equal to zero.

Graphing the Quadratic Expression

To visualize the behavior of the quadratic expression, we can graph it on a coordinate plane. The graph of the quadratic expression is a parabola that opens downward, as indicated by the negative coefficient of the $x^2$ term.

Determining the Intervals of Positivity

To determine the intervals of positivity, we need to examine the behavior of the quadratic expression between its roots. We can do this by selecting a test point within each interval and evaluating the expression at that point.

Interval 1: $(-\infty, -3)$

To determine the behavior of the quadratic expression in the interval $(-\infty, -3)$, we can select a test point, such as $x = -4$. Evaluating the expression at this point, we get $-(-4)^2 - 5(-4) - 6 = -16 + 20 - 6 = -2$. Since the expression is negative at this point, we can conclude that the quadratic expression is negative in the interval $(-\infty, -3)$.

Interval 2: $(-3, -2)$

To determine the behavior of the quadratic expression in the interval $(-3, -2)$, we can select a test point, such as $x = -2.5$. Evaluating the expression at this point, we get $-(-2.5)^2 - 5(-2.5) - 6 = -6.25 + 12.5 - 6 = 0.25$. Since the expression is positive at this point, we can conclude that the quadratic expression is positive in the interval $(-3, -2)$.

Interval 3: $(-2, \infty)$

To determine the behavior of the quadratic expression in the interval $(-2, \infty)$, we can select a test point, such as $x = -1$. Evaluating the expression at this point, we get $-(-1)^2 - 5(-1) - 6 = -1 + 5 - 6 = -2$. Since the expression is negative at this point, we can conclude that the quadratic expression is negative in the interval $(-2, \infty)$.

Conclusion

In conclusion, the quadratic expression $-x^2 - 5x - 6$ is positive in the interval $(-3, -2)$. This interval represents the values of $x$ for which the expression is greater than zero.

Final Answer

Q: What is the main goal of solving the inequality $-x^2 - 5x - 6 < 0$?

A: The main goal of solving the inequality $-x^2 - 5x - 6 < 0$ is to determine the values of the variable $x$ for which the expression $-x^2 - 5x - 6$ is positive.

Q: What is the first step in solving the inequality $-x^2 - 5x - 6 < 0$?

A: The first step in solving the inequality $-x^2 - 5x - 6 < 0$ is to factor the quadratic expression $-x^2 - 5x - 6$.

Q: How do you factor the quadratic expression $-x^2 - 5x - 6$?

A: To factor the quadratic expression $-x^2 - 5x - 6$, we need to find two numbers whose product is $-6$ and whose sum is $-5$. These numbers are $-3$ and $-2$, as their product is $-6$ and their sum is $-5$. Therefore, we can write the quadratic expression as $-(x + 3)(x + 2)$.

Q: What are the roots of the quadratic expression $-x^2 - 5x - 6$?

A: The roots of the quadratic expression $-x^2 - 5x - 6$ are the values of $x$ that make the expression equal to zero. In this case, the roots are $x = -3$ and $x = -2$.

Q: How do you determine the intervals of positivity for the quadratic expression $-x^2 - 5x - 6$?

A: To determine the intervals of positivity for the quadratic expression $-x^2 - 5x - 6$, we need to examine the behavior of the expression between its roots. We can do this by selecting a test point within each interval and evaluating the expression at that point.

Q: What is the interval of positivity for the quadratic expression $-x^2 - 5x - 6$?

A: The interval of positivity for the quadratic expression $-x^2 - 5x - 6$ is $(-3, -2)$.

Q: Why is it important to understand the behavior of the quadratic expression $-x^2 - 5x - 6$?

A: Understanding the behavior of the quadratic expression $-x^2 - 5x - 6$ is important because it allows us to determine the values of the variable $x$ for which the expression is positive. This is useful in a variety of applications, such as optimization problems and data analysis.

Q: How can you apply the concept of solving the inequality $-x^2 - 5x - 6 < 0$ to real-world problems?

A: The concept of solving the inequality $-x^2 - 5x - 6 < 0$ can be applied to real-world problems in a variety of ways. For example, it can be used to determine the optimal values of a variable in an optimization problem, or to analyze the behavior of a system in a data analysis problem.

Q: What are some common mistakes to avoid when solving the inequality $-x^2 - 5x - 6 < 0$?

A: Some common mistakes to avoid when solving the inequality $-x^2 - 5x - 6 < 0$ include:

• Not factoring the quadratic expression correctly
• Not identifying the roots of the quadratic expression correctly
• Not determining the intervals of positivity correctly
• Not applying the concept of solving the inequality to real-world problems correctly

Q: How can you check your work when solving the inequality $-x^2 - 5x - 6 < 0$?

A: To check your work when solving the inequality $-x^2 - 5x - 6 < 0$, you can:

• Verify that the quadratic expression is factored correctly
• Verify that the roots of the quadratic expression are identified correctly
• Verify that the intervals of positivity are determined correctly
• Verify that the concept of solving the inequality is applied correctly to real-world problems.