Math Problem Statement
solve the inequality
Solution
Let's solve each inequality analytically.
Given inequalities:
- (a)
- (b)
Step-by-Step Solution
1. Factor the quadratic expression:
The expression can be factored as follows: Therefore, the factored form is .
2. Find the roots of the quadratic equation
Setting : Thus, the roots are:
Solution for (a):
The inequality is .
-
Identify the intervals based on the roots:
-
Choose a test point in each interval to determine the sign:
- For , choose :
- For , choose :
- For , choose :
-
Thus, the sign of the expression in each interval is:
- : Positive
- : Negative
- : Positive
-
Solution: or .
-
In interval notation: .
Solution for (b):
The inequality is .
-
From the interval analysis above, the expression is negative only in the interval:
- .
-
Solution: .
-
In interval notation: .
Final Answers
- (a)
- (b)
Would you like further details or have any questions?
5 Related Questions
- How do you determine the intervals to test the sign of a quadratic expression?
- What is the significance of roots in solving inequalities?
- How would the solution change if the inequality sign was reversed for part (a)?
- Can you explain the graphical interpretation of the inequality solutions?
- How would you solve inequalities with non-factorable quadratic expressions?
Tip:
Always test points in each interval determined by the roots to ensure the correct sign when solving inequalities.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Inequalities
Factoring
Formulas
Quadratic factoring: ax^2 + bx + c = 0
Sign testing in intervals based on roots
Theorems
Factor theorem
Quadratic inequality solution method
Suitable Grade Level
Grades 9-11