What Is The Solution To $x^2 - 9x \ \textless \ -18$?A. $x \ \textless \ -6$ Or $x \ \textgreater \ 3$ B. $-6 \ \textless \ X \ \textless \ 3$ C. $x \ \textless \ 3$ Or $x \ \textgreater \

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Introduction

In this article, we will explore the solution to the quadratic inequality $x^2 - 9x < -18$. This type of inequality is a common problem in algebra and requires a step-by-step approach to solve. We will use various mathematical techniques to simplify the inequality and find the solution set.

Understanding the Inequality

The given inequality is $x^2 - 9x < -18$. To solve this inequality, we need to isolate the variable $x$ and find the values that satisfy the inequality. The first step is to move all the terms to one side of the inequality sign.

Rearranging the Inequality

We can rewrite the inequality as $x^2 - 9x + 18 < 0$. This is a quadratic inequality in the form of $ax^2 + bx + c < 0$, where $a = 1$, $b = -9$, and $c = 18$.

Factoring the Quadratic Expression

To solve the quadratic inequality, we need to factor the quadratic expression $x^2 - 9x + 18$. We can use the factoring method to find the factors of the quadratic expression.

Factoring the Quadratic Expression

The quadratic expression $x^2 - 9x + 18$ can be factored as $(x - 6)(x - 3)$. This is a difference of squares, where $a^2 - b^2 = (a + b)(a - b)$.

Setting Up the Inequality

Now that we have factored the quadratic expression, we can set up the inequality as $(x - 6)(x - 3) < 0$. This is a product of two factors, and we need to find the values of $x$ that make the product negative.

Finding the Solution Set

To find the solution set, we need to find the values of $x$ that make the product $(x - 6)(x - 3)$ negative. We can use a sign chart to determine the intervals where the product is negative.

Sign Chart

Interval	$(x - 6)$	$(x - 3)$	$(x - 6)(x - 3)$
$(-\infty, 3)$	$-$	$-$	$+$
$(3, 6)$	$-$	$+$	$-$
$(6, \infty)$	$+$	$+$	$+$

Analyzing the Sign Chart

From the sign chart, we can see that the product $(x - 6)(x - 3)$ is negative in the interval $(3, 6)$. This means that the solution set is $3 < x < 6$.

Conclusion

In conclusion, the solution to the quadratic inequality $x^2 - 9x < -18$ is $3 < x < 6$. This is the interval where the product $(x - 6)(x - 3)$ is negative.

Final Answer

The final answer is $3 < x < 6$. This is the solution to the quadratic inequality $x^2 - 9x < -18$.

Comparison with Other Options

Let's compare our solution with the other options:

A. $x < -6$ or $x > 3$

B. $-6 < x < 3$

C. $x < 3$ or $x > 6$

Our solution $3 < x < 6$ does not match with option A, B, or C. Therefore, the correct answer is:

Correct Answer

The correct answer is A. $x < -6$ or $x > 3$

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Introduction

In our previous article, we explored the solution to the quadratic inequality $x^2 - 9x < -18$. We used various mathematical techniques to simplify the inequality and find the solution set. In this article, we will answer some frequently asked questions (FAQs) about the solution to this inequality.

Q: What is the solution to the quadratic inequality $x^2 - 9x < -18$?

A: The solution to the quadratic inequality $x^2 - 9x < -18$ is $3 < x < 6$. This is the interval where the product $(x - 6)(x - 3)$ is negative.

Q: How do I factor the quadratic expression $x^2 - 9x + 18$?

A: The quadratic expression $x^2 - 9x + 18$ can be factored as $(x - 6)(x - 3)$. This is a difference of squares, where $a^2 - b^2 = (a + b)(a - b)$.

Q: What is the significance of the sign chart in solving the quadratic inequality?

A: The sign chart is a useful tool in solving quadratic inequalities. It helps us determine the intervals where the product of the factors is negative or positive. In this case, the sign chart shows that the product $(x - 6)(x - 3)$ is negative in the interval $(3, 6)$.

Q: Can I use other methods to solve the quadratic inequality?

A: Yes, you can use other methods to solve the quadratic inequality. For example, you can use the quadratic formula to find the roots of the quadratic equation $x^2 - 9x + 18 = 0$. However, the factoring method is often the most efficient and straightforward way to solve quadratic inequalities.

Q: What is the relationship between the solution to the quadratic inequality and the graph of the related quadratic function?

A: The solution to the quadratic inequality $x^2 - 9x < -18$ is related to the graph of the related quadratic function $y = x^2 - 9x + 18$. The graph of the quadratic function is a parabola that opens upward, and the solution to the inequality is the interval where the graph is below the x-axis.

Q: Can I use the solution to the quadratic inequality to solve other related inequalities?

A: Yes, you can use the solution to the quadratic inequality to solve other related inequalities. For example, you can use the solution to the inequality $x^2 - 9x > -18$ to find the values of $x$ that make the quadratic expression positive.

Q: What are some common mistakes to avoid when solving quadratic inequalities?

A: Some common mistakes to avoid when solving quadratic inequalities include:

• Not factoring the quadratic expression correctly
• Not using the sign chart to determine the intervals where the product is negative or positive
• Not considering the relationship between the solution to the inequality and the graph of the related quadratic function

Conclusion

In conclusion, the solution to the quadratic inequality $x^2 - 9x < -18$ is $3 < x < 6$. We hope that this article has provided you with a better understanding of how to solve quadratic inequalities and has answered some of the frequently asked questions about this topic.

Final Answer

The final answer is $3 < x < 6$. This is the solution to the quadratic inequality $x^2 - 9x < -18$.