Math Problem Statement
Solve the inequality. (Enter your answer using interval notation.)
−2x2 + 2x ≤ −25
Solution
Let's solve the inequality step by step:
The given inequality is:
Step 1: Move all terms to one side.
Add 25 to both sides to bring everything to one side:
Step 2: Divide through by -2.
To simplify the expression, divide the entire inequality by -2. Remember, dividing by a negative number reverses the inequality sign:
Step 3: Solve the quadratic equation.
At this point, solving the quadratic equation would involve the discriminant:
This expression simplifies to find the roots. However, it looks a bit complicated and likely involves a non-factorable quadratic.
Let me check this more carefullyThe solution to the inequality is:
This means that the inequality holds for values less than or equal to approximately , and for values greater than or equal to approximately .
Do you want more details or have any questions?
Here are 5 related questions for further exploration:
- How would the solution change if the inequality were strict instead of ?
- What are the exact roots of the quadratic equation?
- How do we graph the inequality on a number line?
- What is the role of the discriminant in solving quadratic inequalities?
- Can we apply this approach to solving cubic inequalities?
Tip: Always remember that dividing an inequality by a negative number reverses the inequality sign!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Inequalities
Formulas
Quadratic equation formula: ax^2 + bx + c = 0
Discriminant: Δ = b^2 - 4ac
Interval notation
Theorems
Quadratic Formula
Properties of Inequalities
Suitable Grade Level
Grades 9-11