Solve The Inequality: $u^2 \ \textless \ -3u - 10$Interval Notation Solution: □ \square □ Options: - No Solution

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Introduction

In this article, we will delve into the world of inequalities and explore the process of solving a quadratic inequality. The given inequality is $u^2 \ \textless \ -3u - 10$, and our goal is to find the solution set in interval notation. We will break down the solution step by step, using algebraic manipulations and logical reasoning to arrive at the final answer.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression on one side of the inequality sign and a constant on the other side. In this case, we have $u^2$ on the left-hand side and $-3u - 10$ on the right-hand side. The inequality sign is "less than," which means we are looking for values of $u$ that make the left-hand side smaller than the right-hand side.

Step 1: Move All Terms to One Side

To solve the inequality, we need to move all the terms to one side of the inequality sign. We can do this by adding $3u + 10$ to both sides of the inequality. This gives us:

$$u^2 + 3u + 10 \ \textless \ 0$$

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the left-hand side. We can do this by finding two numbers whose product is $10$ and whose sum is $3$. These numbers are $2$ and $5$, so we can factor the quadratic expression as:

$$(u + 2)(u + 5) \ \textless \ 0$$

Step 3: Find the Critical Points

The critical points are the values of $u$ that make the quadratic expression equal to zero. In this case, the critical points are $u = -2$ and $u = -5$. These points divide the number line into three intervals: $(-\infty, -5)$, $(-5, -2)$, and $(-2, \infty)$.

Step 4: Test Each Interval

To determine which intervals satisfy the inequality, we need to test each interval by plugging in a test value from each interval. Let's choose $u = -6$ from the interval $(-\infty, -5)$, $u = -3$ from the interval $(-5, -2)$, and $u = -1$ from the interval $(-2, \infty)$.

Step 5: Evaluate the Inequality for Each Interval

Evaluating the inequality for each interval, we get:

• For $u = -6$, we have $(u + 2)(u + 5) = (-6 + 2)(-6 + 5) = (-4)(-1) = 4$, which is positive.
• For $u = -3$, we have $(u + 2)(u + 5) = (-3 + 2)(-3 + 5) = (-1)(2) = -2$, which is negative.
• For $u = -1$, we have $(u + 2)(u + 5) = (-1 + 2)(-1 + 5) = (1)(4) = 4$, which is positive.

Step 6: Determine the Solution Set

Based on the results of the interval testing, we can determine that the solution set is the interval $(-5, -2)$.

Conclusion

In conclusion, the solution to the inequality $u^2 \ \textless \ -3u - 10$ is the interval $(-5, -2)$. This means that any value of $u$ between $-5$ and $-2$ (exclusive) satisfies the inequality.

Final Answer

The final answer is $\boxed{(-5, -2)}$.

$$

Introduction

In this article, we will delve into the world of inequalities and explore the process of solving a quadratic inequality. The given inequality is $u^2 \ \textless \ -3u - 10$, and our goal is to find the solution set in interval notation. We will break down the solution step by step, using algebraic manipulations and logical reasoning to arrive at the final answer.

Understanding the Inequality

The given inequality is a quadratic inequality, which means it involves a quadratic expression on one side of the inequality sign and a constant on the other side. In this case, we have $u^2$ on the left-hand side and $-3u - 10$ on the right-hand side. The inequality sign is "less than," which means we are looking for values of $u$ that make the left-hand side smaller than the right-hand side.

Step 1: Move All Terms to One Side

To solve the inequality, we need to move all the terms to one side of the inequality sign. We can do this by adding $3u + 10$ to both sides of the inequality. This gives us:

$$u^2 + 3u + 10 \ \textless \ 0$$

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the left-hand side. We can do this by finding two numbers whose product is $10$ and whose sum is $3$. These numbers are $2$ and $5$, so we can factor the quadratic expression as:

$$(u + 2)(u + 5) \ \textless \ 0$$

Step 3: Find the Critical Points

The critical points are the values of $u$ that make the quadratic expression equal to zero. In this case, the critical points are $u = -2$ and $u = -5$. These points divide the number line into three intervals: $(-\infty, -5)$, $(-5, -2)$, and $(-2, \infty)$.

Step 4: Test Each Interval

To determine which intervals satisfy the inequality, we need to test each interval by plugging in a test value from each interval. Let's choose $u = -6$ from the interval $(-\infty, -5)$, $u = -3$ from the interval $(-5, -2)$, and $u = -1$ from the interval $(-2, \infty)$.

Step 5: Evaluate the Inequality for Each Interval

Evaluating the inequality for each interval, we get:

• For $u = -6$, we have $(u + 2)(u + 5) = (-6 + 2)(-6 + 5) = (-4)(-1) = 4$, which is positive.
• For $u = -3$, we have $(u + 2)(u + 5) = (-3 + 2)(-3 + 5) = (-1)(2) = -2$, which is negative.
• For $u = -1$, we have $(u + 2)(u + 5) = (-1 + 2)(-1 + 5) = (1)(4) = 4$, which is positive.

Step 6: Determine the Solution Set

Based on the results of the interval testing, we can determine that the solution set is the interval $(-5, -2)$.

Conclusion

In conclusion, the solution to the inequality $u^2 \ \textless \ -3u - 10$ is the interval $(-5, -2)$. This means that any value of $u$ between $-5$ and $-2$ (exclusive) satisfies the inequality.

Final Answer

The final answer is $\boxed{(-5, -2)}$.

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Q&A: Solving the Inequality $u^2 \ \textless \ -3u - 10$

Q: What is the first step in solving the inequality $u^2 \ \textless \ -3u - 10$?

A: The first step is to move all the terms to one side of the inequality sign by adding $3u + 10$ to both sides of the inequality.

Q: How do we factor the quadratic expression on the left-hand side of the inequality?

A: We can factor the quadratic expression by finding two numbers whose product is $10$ and whose sum is $3$. These numbers are $2$ and $5$, so we can factor the quadratic expression as $(u + 2)(u + 5)$.

Q: What are the critical points of the inequality?

A: The critical points are the values of $u$ that make the quadratic expression equal to zero. In this case, the critical points are $u = -2$ and $u = -5$.

Q: How do we determine which intervals satisfy the inequality?

A: We can determine which intervals satisfy the inequality by testing each interval by plugging in a test value from each interval.

Q: What is the solution set of the inequality?

A: The solution set of the inequality is the interval $(-5, -2)$.

Q: What does the solution set mean?

A: The solution set means that any value of $u$ between $-5$ and $-2$ (exclusive) satisfies the inequality.

Q: How do we evaluate the inequality for each interval?

A: We can evaluate the inequality for each interval by plugging in a test value from each interval and checking if the inequality is true or false.

Q: What is the final answer to the inequality?

A: The final answer to the inequality is $\boxed{(-5, -2)}$.

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Additional Resources

• For more information on solving quadratic inequalities, see the article "Solving Quadratic Inequalities".
• For more practice problems on solving quadratic inequalities, see the article "Practice Problems: Solving Quadratic Inequalities".
• For more resources on algebra and mathematics, see the article "Resources: Algebra and Mathematics".