Inequalities

Given two numbers $a$ and $b$ , we write $a < b$ ( $a$ is less than $b$ ) or equivalently $b > a$ ( $b$ is greater than $a$ ) if $b - a$ is positive. Geometrically $a < b$ means $a$ lies to the left of $b$ on the number line (see the following figure).

The symbol $a \leq b$ means either $a < b$ or $a = b$ and
$a < b \leq c$ means $a < b$ and $b \leq c$ .

Illustration for Inequalities — a < b geometrically means a lies to the left of b on the number line.

The signs $<$ and $>$ are called inequality symbols and satisfy the following properties:

If 
  a
  ≠
  b
 then 
  a
  <
  b
 or 
  a
  >
  b
.
If 
  a
  >
  b
 and 
  b
  >
  c
 then 
  a
  >
  c
.
If 
  a
  >
  b
 then 
  a
  +
  c
  >
  b
  +
  c
 (and 
  a
  −
  c
  >
  b
  −
  c
) for every 
  c
 (if we add a positive or negative number to both sides of an inequality, the direction of the inequality will be preserved).
If 
  a
  >
  b
 and 
  c
  >
  d
, then 
  a
  +
  c
  >
  b
  +
  d
 (Inequalities with the same directions can be added).
If 
  a
  >
  b
 and 
  c
  >
  0
 then 
  a
  c
  >
  b
  c
 (If we multiply or divide both sides of an inequality by a positive number, the direction of the inequality will be preserved).
If 
  a
  >
  b
 and 
  c
  <
  0
 then 
  a
  c
  <
  b
  c
 (If we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality direction).
If 
  a
 and 
  b
 are both positive or both negative and 
  a
  <
  b
 then 
  
    
      1
      a
    
  
  >
  
    
      1
      b
    
  
.
If 
  a
  <
  0
  <
  b
, then 
  
    
      1
      a
    
  
  <
  0
  <
  
    
      1
      b
    
  
.
If 
  a
  ≠
  0
, 
  
    a
    
      2
    
  
  >
  0
.

The above properties remain true, if we replace $>$ by $\geq$ and $<$ by $\leq$ .

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