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
a2+b2≥2aba^2 + b^2 \geq 2ab
Equality holds if a=ba = b.
Arithmetic Mean ≥\geq Geometric Mean ≥\geq Harmonic Mean: AM≥GM≥HM\text{AM} \geq \text{GM} \geq \text{HM}
a2+b2+c2≥ab+bc+caa^2 + b^2 + c^2 \geq ab + bc + ca
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.
If a+b=2a + b = 2, then a4+b4≥2a^4 + b^4 \geq 2.
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}
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.