Math Problem Statement
Solution
The problem in the image requires finding the solution to the inequality and then expressing the solution in both Set-Builder Notation and Interval Notation.
Step 1: Solve the inequality
We first need to solve the corresponding quadratic equation: We can solve this quadratic equation using the quadratic formula: where , , and .
So, we have two solutions:
Step 2: Determine the sign of
The quadratic opens upwards because the coefficient of is positive. The roots divide the number line into three intervals:
We test the sign of the quadratic expression in each interval:
- For , pick :
- For , pick :
- For , pick :
So, the quadratic expression is positive (or zero) on the intervals and .
Step 3: Write the solution in Set-Builder and Interval Notation
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Set-Builder Notation:
-
Interval Notation:
Would you like more details on any step? Here are some related questions:
- How do you determine the sign of a quadratic expression in different intervals?
- What is the significance of the direction in which a parabola opens?
- How does the quadratic formula work, and why is it applicable here?
- Can we solve the quadratic inequality using a different method, like factoring?
- What is the difference between Set-Builder Notation and Interval Notation?
Tip: When solving inequalities, always consider the intervals between the roots and test them to understand the behavior of the quadratic function.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Inequalities
Set-Builder Notation
Interval Notation
Formulas
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Quadratic Formula
Sign Analysis for Quadratic Expressions
Suitable Grade Level
Grades 9-11