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:

2x2+2x25-2x^2 + 2x \leq -25

Step 1: Move all terms to one side.

Add 25 to both sides to bring everything to one side:

2x2+2x+250-2x^2 + 2x + 25 \leq 0

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:

x2x2520x^2 - x - \frac{25}{2} \geq 0

Step 3: Solve the quadratic equation.

At this point, solving the quadratic equation would involve the discriminant:

x=(1)±(1)24(1)(252)2(1)x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\frac{25}{2})}}{2(1)}

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 x2x2520x^2 - x - \frac{25}{2} \geq 0 is:

x(,3.07][4.07,)x \in (-\infty, -3.07] \cup [4.07, \infty)

This means that the inequality holds for xx values less than or equal to approximately 3.07-3.07, and for xx values greater than or equal to approximately 4.074.07.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

  1. How would the solution change if the inequality were strict >> instead of \geq?
  2. What are the exact roots of the quadratic equation?
  3. How do we graph the inequality on a number line?
  4. What is the role of the discriminant in solving quadratic inequalities?
  5. Can we apply this approach to solving cubic inequalities?

Tip: Always remember that dividing an inequality by a negative number reverses the inequality sign!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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