Understanding Inequalities: A Comprehensive Guide

Understanding Inequalities: A Comprehensive Guide

Understanding Inequalities: A Comprehensive Guide

Introduction to Inequalities

For any two real numbers aa and bb, one of three possibilities always holds:
  • a<ba < b: aa is less than bb
  • a=ba = b: aa is equal to bb
  • a>ba > b: aa is greater than bb
An inequality involves a comparison between two values using inequality signs, such as >>, <<, ≥\geq, or ≤\leq. These comparisons can also incorporate equations, leading to solutions that define a range of values satisfying the inequality. 

Inequality Signs and Their Meanings

  • >>: Greater than
  • <<: Less than
  • ≥\geq: Greater than or equal to
  • ≤\leq: Less than or equal to

Properties of Inequalities

 

1. Trichotomy Property

For any two real numbers aa and bb, exactly one of the following is true:
  • a<ba < b
  • a=ba = b
  • a>ba > b

2. Transitive Properties of Inequalities

  • If a<ba < b and b<cb < c, then a<ca < c.
  • If a>ba > b and b>cb > c, then a>ca > c.

3. Properties of Addition and Subtraction

An equal quantity can be added to or subtracted from both sides of an inequality without altering the inequality:
  • If a<ba < b, then a+c<b+ca + c < b + c.
  • If a>ba > b, then a+c>b+ca + c > b + c.
  • If a<ba < b, then a−c<b−ca – c < b – c.
  • If a>ba > b, then a−c>b−ca – c > b – c.

4. Properties of Multiplication and Division

Positive Multiplication/Division

If both sides of an inequality are multiplied or divided by a positive number, the inequality remains unchanged:
  • If a<ba < b and c>0c > 0, then ac<bcac < bc and ac<bc\frac{a}{c} < \frac{b}{c}.
  • If a>ba > b and c>0c > 0, then ac>bcac > bc and ac>bc\frac{a}{c} > \frac{b}{c}.

Negative Multiplication/Division

If both sides of an inequality are multiplied or divided by a negative number, the inequality is reversed:
  • If a<ba < b and c<0c < 0, then ac>bcac > bc and ac>bc\frac{a}{c} > \frac{b}{c}.
  • If a>ba > b and c<0c < 0, then ac<bcac < bc and ac<bc\frac{a}{c} < \frac{b}{c}.

Important Inequalities

  1. a2+b2≥2aba^2 + b^2 \geq 2ab Equality holds if a=ba = b.
  2. Arithmetic Mean ≥\geq Geometric Mean ≥\geq Harmonic Mean: AM≥GM≥HM\text{AM} \geq \text{GM} \geq \text{HM}
  3. a2+b2+c2≥ab+bc+caa^2 + b^2 + c^2 \geq ab + bc + ca
  4. a3+b3≥ab(a+b)a^3 + b^3 \geq ab(a+b) if a>0a > 0 and b>0b > 0, with equality only if a=ba = b.
  5. If a+b=2a + b = 2, then a4+b4≥2a^4 + b^4 \geq 2.
  6. 2n>n22^n > n^2 for n≥5n \geq 5.

Modulus or Absolute Value

The absolute value of a real number is its distance from zero on the number line, disregarding its sign. For any real number aa, the absolute value, denoted by ∣a∣|a|, is defined as:∣a∣={aif a≥0−aif a<0|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}

Properties of Modulus

  • ∣a∣≥0|a| \geq 0 for all real numbers aa.
  • ∣a∣=0|a| = 0 if and only if a=0a = 0.
  • ∣ab∣=∣a∣⋅∣b∣|ab| = |a| \cdot |b|.
  • ∣a+b∣≤∣a∣+∣b∣|a + b| \leq |a| + |b| (Triangle Inequality).
  • ∣a−b∣≥∣∣a∣−∣b∣∣|a – b| \geq \big| |a| – |b| \big|.

Solved Examples

Example 1: Solve for xx if 2x+5>152x + 5 > 15. Solution: 2x>10⇒x>52x > 10 \quad \Rightarrow \quad x > 5Example 2: Find the range of xx for which ∣x−3∣≤4|x – 3| \leq 4. Solution: −4≤x−3≤4⇒−1≤x≤7-4 \leq x – 3 \leq 4 \quad \Rightarrow \quad -1 \leq x \leq 7

Conclusion

Inequalities are a fundamental concept in mathematics, crucial for solving a variety of real-world problems and advanced equations. Understanding their properties, types, and solving techniques equips you with essential analytical skills. Mastery of inequalities not only enhances your problem-solving abilities but also lays a strong foundation for more complex mathematical concepts.
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