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Aptitude Topics

Inequalities

An inequality is a mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥. Solving an inequality gives a range of values rather than a single answer.

Fundamental Principles

Linear Inequality

An inequality involving a linear expression, e.g., 2x + 3 < 7. Its solution is an interval on the number line.

Quadratic Inequality

An inequality involving a quadratic expression, e.g., x² - 5x + 6 > 0. Its solution requires finding roots and analyzing sign patterns.

Absolute Value Inequality

|x| < a means -a < x < a. |x| > a means x < -a or x > a.

Essential Formulation Tips

  • When multiplying or dividing by a negative number, flip the inequality sign.
  • For quadratic inequalities, find roots first, then test intervals.
  • Graph on a number line to visualize the solution set.
  • Absolute value splits into two cases: positive and negative.

Shortcut Execution Techniques

  • ax + b < c → x < (c-b)/a if a > 0; reverse if a < 0.
  • For x² > a²: x > a or x < -a.
  • For x² < a²: -a < x < a.
  • AM ≥ GM: (a+b)/2 ≥ √(ab) for non-negative a, b.

Contextual Inquiries (FAQs)

Q: Does the inequality sign always stay the same?

A: No. When multiplying or dividing both sides by a negative number, the sign flips.

Q: How do you solve a quadratic inequality?

A: Find the roots, mark them on a number line, and test each interval to determine where the inequality holds.

Q: What does |x - 3| < 2 mean geometrically?

A: It means x is within 2 units of 3, i.e., 1 < x < 5.