Math Problem Statement
Solution
To solve the inequality given in the image:
we need to analyze the inequality by finding the intervals where this expression is positive.
Step 1: Identify the Zeros
The inequality is a product of three terms. We start by finding the zeros of each term:
- implies
- implies
- implies
These zeros are , , and .
Step 2: Analyze Intervals
Using these zeros, we divide the real line into intervals:
For each interval, we test a point within the interval to see if the expression is positive or negative.
Step 3: Sign Analysis
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Interval : Pick . The product (positive).
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Interval : Pick . The product (negative).
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Interval : Pick . The product (negative).
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Interval : Pick . The product (positive).
Step 4: Solution
The intervals where (positive) are:
Thus, the solution to the inequality is:
Let me know if you need further details or explanations on any of the steps.
Here are five related questions for further exploration:
- How would the solution change if the inequality were instead of ?
- What would the solution be if the inequality were ?
- Can this polynomial inequality be solved graphically, and how?
- How do changes to the coefficients affect the intervals where the inequality holds?
- How can interval notation help represent the solution clearly in complex inequalities?
Tip: For polynomial inequalities, always find the zeros first, then use interval testing to determine the sign of the expression within each interval.
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Math Problem Analysis
Mathematical Concepts
Polynomial Inequalities
Interval Testing
Real Zeros
Formulas
Product of factors p(x) = x^2(x + 4)(x - 1)
Theorems
Sign Testing in Intervals for Polynomial Inequalities
Suitable Grade Level
Grades 10-12