Math Problem Statement
(x + 9)2(x + 7) > 0
Solution
To solve the inequality , follow these steps:
Step 1: Identify the critical points
We first look for the values of that make the expression equal to 0. The expression is equal to 0 when:
- From , we get .
- From , we get .
Thus, the critical points are and .
Step 2: Analyze the sign of the expression in intervals
We will now divide the real number line into intervals based on these critical points and determine the sign of the expression in each interval.
The intervals are:
Step 3: Test the sign in each interval
We test a point in each interval:
-
For (let ):
So, the expression is negative in this interval. -
For (let ):
So, the expression is negative in this interval. -
For (let ):
So, the expression is positive in this interval.
Step 4: Conclusion
From the sign analysis, we find that the expression is:
- Negative for
- Negative for
- Positive for
Since the inequality is , we want the expression to be positive. Therefore, the solution is:
Step 5: Exclude the critical points
At , , and at , . At these points, the expression equals 0, not greater than 0, so we exclude them from the solution.
Thus, the solution to the inequality is:
Final Answer:
Would you like further clarification on any of the steps? Here are some related questions you might find interesting:
- What does the square term imply for the solution?
- How do we solve quadratic inequalities like ?
- What is the behavior of the function near its critical points?
- How can you graphically represent the solution to this inequality?
- What would happen if we had a cubic term instead of a quadratic one?
Tip: Always check the behavior of the expression at critical points, as they often represent boundaries where the sign of the expression changes!
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Math Problem Analysis
Mathematical Concepts
Inequalities
Quadratic Equations
Sign Analysis
Critical Points
Formulas
(x + 9)² = 0
(x + 7) = 0
Theorems
Sign analysis method
Behavior of quadratic functions near critical points
Suitable Grade Level
Grades 9-11