Inequalities

• Wavy curve method for solving (x−a)(x−b)(x−c) > 0
• Arrange the critical points in ascending order.
  $$a<b<c$$
• Number line representation:
  
  
  
• 
• $$(x-a)(x-b)(x-c) > 0 \Rightarrow x \in (a,b)\cup(c,\infty)$$

• The modulus of x, |x| equals the maximum of x and –x
• 
• For any two real numbers 'a' and 'b', 

                                  |a|+|b| ≥ |a+b|
                                    |a|-|b| ≤ |a-b|                                 

                                   |a.b| = |a| |b|

• If |x| ≤ k then the value of x lies between –k and k, or –k ≤ x ≤ k
• If |x| ≥ k then x ≥ k or x ≤ -k

• For any real number 'a', $$a^2$$ = $$|a|^2$$

• If a$$x^{2}$$+bx+c < 0 then a(x-m)(x-n) < 0.
  If a>0 and n>m, then m < x < n
  if a<0 and n>m, then x < m and x > n
• If a$$x^{2}$$+bx+c > 0 then a(x-m)(x-n) > 0.
  If a>0 and n>m, then x < m and x > n
  if a<0 and n>m, then m < x < n
• If a$$x^{2}$$+bx+c > 0 but m = n, then the value of x exists for all values, except x is equal to m, i.e., x < m and x > m but x ≠ m
• If a, x, b are positive, ax > b => x > $$\dfrac{b}{a}$$ and ax < b => x < $$\dfrac{b}{a}$$

• For any three real numbers X, Y and Z; if X > Y then X+Z > Y+Z

• If X > Y and

  1. Z is positive, then XZ > YZ
  2. Z is negative, then XZ < YZ
  3. If X and Y are of the same sign, $$\dfrac{1}{X}$$ < $$\dfrac{1}{Y}$$
  4. If X and Y are of different signs, $$\dfrac{1}{X}$$ > $$\dfrac{1}{Y}$$
  5. Squaring rule: If X, Y > 0 and X > Y then X² > Y² 
     
  6. If 0 < X < 1: X² < X < √X — behaviour of fractions under powers
     
  7. If X > 1: X² > X > √X

Prerequisites

The logarithm $$\log_a x$$ is defined only when $$x > 0$$, $$a > 0$$ and $$a \neq 1$$.

Before solving any logarithmic inequality, find the domain of every logarithmic expression first. Solve the inequality, then take the intersection of the solution with the domain.

Monotonicity: The core idea

The direction of a logarithmic inequality depends entirely on the base.

If $$a > 1$$, then $$\log_a x$$ is an increasing function:
$$x_1 < x_2 \iff \log_a x_1 < \log_a x_2$$
The inequality sign is preserved.

If $$0 < a < 1$$, then $$\log_a x$$ is a decreasing function:
$$x_1 < x_2 \iff \log_a x_1 > \log_a x_2$$
The inequality sign is reversed.

Relationship between AM, GM and HM for two numbers a and b,

• G.M=$$\sqrt{AM \times HM}$$
• A.M ≥ G.M ≥ H.M
• The equality holds true if and only if all the terms are equal. (AM = GM = HM)