Math Problem Statement

Solve quadratic and rational inequalities, e.g. x^2 - x - 12 > 0, (x+2)/(x-4) - (x-4)/(x+2) > 0.

Solution

The image you've uploaded contains a set of quadratic inequalities and rational inequalities written in Russian. Here's a breakdown of the types of problems:

  1. Quadratic Inequalities:

    • Problems where you need to solve inequalities involving quadratic expressions.
    • Example: x2x12>0x^2 - x - 12 > 0 or x26x+9<0x^2 - 6x + 9 < 0.
  2. Rational Inequalities:

    • Problems where you need to solve inequalities involving rational expressions.
    • Example: x+2x4x4x+2>0\frac{x + 2}{x - 4} - \frac{x - 4}{x + 2} > 0.

Would you like to start with a specific problem? Let me know which one you'd like to solve, or I can walk you through an example step-by-step!

Here are five related questions that expand on quadratic and rational inequalities:

  1. How do you solve a quadratic inequality like x2x12>0x^2 - x - 12 > 0?
  2. What methods can be used to solve rational inequalities?
  3. How can you graph the solution set of a quadratic inequality on a number line?
  4. How do you identify the critical points for inequalities involving polynomials?
  5. What is the role of factoring in solving quadratic inequalities?

Tip: Always factor quadratic expressions first to find the critical points before solving inequalities.

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Math Problem Analysis

Mathematical Concepts

Quadratic Inequalities
Rational Inequalities
Factoring

Formulas

Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a
Critical points for rational inequalities: Solve for when the numerator or denominator equals zero

Theorems

Zero Product Property
Sign Chart Method for inequalities

Suitable Grade Level

Grades 9-12