Math Problem Statement
Solution
The image contains the following inequalities that need to be solved:
(a)
(b)
(c)
Let’s solve these one by one.
(a)
-
Factor the quadratic expression: So, the inequality becomes:
-
Identify the critical points: and .
-
Test intervals:
- For , both factors are negative, so the product is positive.
- For , one factor is positive, and one is negative, so the product is negative.
- For , both factors are positive, so the product is positive.
-
The solution to the inequality is .
(b)
-
Rearrange the inequality:
-
Factor or use the quadratic formula: This gives and .
-
Test intervals:
- For , the quadratic expression is positive.
- For , the quadratic expression is negative.
- For , the quadratic expression is positive.
-
The solution is .
(c)
-
Move all terms to one side: Simplify:
-
Factor the quadratic:
-
Identify critical points: and .
-
Test intervals:
- For , the expression is positive.
- For , the expression is negative.
- For , the expression is positive.
-
The solution is .
Final Solutions:
- (a)
- (b)
- (c)
Would you like further details on any of these steps or explanations?
Here are 5 related questions for further understanding:
- How do you solve inequalities using factorization?
- What is the importance of testing intervals in inequalities?
- How does the quadratic formula help in solving inequalities?
- What happens if you multiply or divide inequalities by a negative number?
- Can inequalities have multiple solution intervals?
Tip: Always test values from different intervals to check the sign of the expression 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
Inequalities
Quadratic Equations
Formulas
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
Theorems
Zero Product Property
Interval Testing Method
Suitable Grade Level
Grades 9-10