Math Problem Statement
Solve this by factoring
Solution
To solve the quadratic inequality by factoring, we need to follow these steps:
1. Factor the quadratic expression
We need to find two numbers that multiply to and add to . The numbers and fit these conditions. Thus, we rewrite the middle term:
Next, we group and factor:
Factor each group:
Now factor out the common factor:
2. Determine the critical points
Set each factor equal to zero to find the critical points:
These critical points divide the number line into intervals: , , and .
3. Test the intervals
We will test the sign of the expression in each interval.
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For , choose :
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For , choose :
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For , choose :
4. Write the solution in interval notation
The inequality is satisfied in the intervals where the product is non-negative, which are and .
5. Set-builder notation
The solution set in set-builder notation is:
Would you like more details or have any questions?
Here are some related questions you might find helpful:
- How do you determine which intervals to test for signs in a quadratic inequality?
- Can all quadratic inequalities be solved by factoring?
- What if the quadratic expression is not factorable? How should you proceed?
- How do the signs of the factors influence the solution of an inequality?
- Why is it important to include the critical points when the inequality is non-strict (e.g., )?
Tip: Always double-check your factorization and the intervals you test to ensure accuracy in solving inequalities.
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Math Problem Analysis
Mathematical Concepts
Quadratic Inequalities
Factoring
Formulas
Quadratic inequality formula ax^2 + bx + c ≥ 0
Theorems
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Suitable Grade Level
Grades 9-12