https://file.aiquickdraw.com/mathbot/file/1730297475747l8o8ovqq.jpg

Let's solve the inequality \(x^2 + 10x + 9 < 0\) step-by-step.

### Step 1: Factor the quadratic expression
We want to factor \(x^2 + 10x + 9\). 

We look for two numbers that multiply to \(9\) (the constant term) and add up to \(10\) (the coefficient of \(x\)). These numbers are \(1\) and \(9\). 

Thus, the factorization is:
\[
x^2 + 10x + 9 = (x + 1)(x + 9)
\]

### Step 2: Set the factors equal to 0 (Find the roots)
We need to determine where the quadratic expression changes sign by solving:
\[
(x + 1)(x + 9) = 0
\]
This gives the roots \(x = -1\) and \(x = -9\).

### Step 3: Analyze the sign changes around the roots
The roots divide the number line into three intervals:  
1. \(x < -9\)  
2. \(-9 < x < -1\)  
3. \(x > -1\)  

We now test a point in each interval to determine the sign of \((x + 1)(x + 9)\) in that interval:

- **Interval 1**: \(x < -9\), try \(x = -10\).  
  \((x + 1)(x + 9) = (-10 + 1)(-10 + 9) = (-9)(-1) = 9\) (positive)

- **Interval 2**: \(-9 < x < -1\), try \(x = -5\).  
  \((x + 1)(x + 9) = (-5 + 1)(-5 + 9) = (-4)(4) = -16\) (negative)

- **Interval 3**: \(x > -1\), try \(x = 0\).  
  \((x + 1)(x + 9) = (0 + 1)(0 + 9) = 1 \cdot 9 = 9\) (positive)

### Step 4: Identify where the expression is negative
We are looking for where the product \((x + 1)(x + 9)\) is **less than 0**. From our sign analysis, the expression is negative in the interval:
\[
-9 < x < -1
\]

### Step 5: Write the solution
The solution to the inequality \(x^2 + 10x + 9 < 0\) is:
\[
-9 < x < -1
\]

### Final Answer
The correct option is:
\[
-9 < x < -1
\]

Do you have any questions or want more details on a specific step? Here are five related questions:

1. How do we solve quadratic inequalities graphically?
2. What changes if the inequality is \(x^2 + 10x + 9 \leq 0\)?
3. How do we solve quadratic inequalities without factoring?
4. What is the difference between strict and non-strict inequalities?
5. How can we check our solution using a graphing tool?

**Tip:** Always test points from different intervals to ensure you correctly capture the sign behavior of the expression!

Solve the inequality algebraically: x^2 + 10x + 9 < 0

Learn how to solve the quadratic inequality x^2 + 10x + 9 < 0 with a step-by-step explanation. This problem explores factoring quadratic expressions and analyzing sign changes around roots. Perfect for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation