Math Problem Statement
Solution
The inequality in the image is:
Let's solve this inequality.
Step 1: Move all terms to one side of the inequality.
Subtract 4 from both sides:
Step 2: Solve the corresponding quadratic equation.
To find the critical points, solve the quadratic equation:
Use the quadratic formula:
where , , and .
So the solutions are:
Step 3: Analyze the intervals.
The critical points divide the number line into three intervals. Test the sign of the quadratic expression in each interval to determine where the inequality holds.
Step 4: Conclusion
- The quadratic expression is greater than or equal to zero in intervals where the test values yield positive results.
- Therefore, the solution to the inequality is:
Would you like a more detailed explanation on any step or have any questions?
Here are 5 related questions that you may find helpful:
- How do you determine the sign of a quadratic expression in different intervals?
- What is the significance of the discriminant in the quadratic formula?
- Can this inequality be solved using a different method?
- How would the solution change if the inequality were strict (i.e., instead of )?
- What are the applications of solving quadratic inequalities?
Tip: Always check the sign of a quadratic expression by testing values in each interval defined by the roots of the equation. This helps in accurately determining the solution set for 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
Quadratic Equations
Formulas
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Sign analysis of a quadratic expression
Roots of a quadratic equation
Suitable Grade Level
Grades 9-12