inequality with absolute values

Recalling that the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( of a real number means on the number line (real axisMathworldPlanetmath) the distance of the point from the origin, we have the following three rules which remove the absolute value signs from an inequality (we use the logical symbol “” for alternativeness ‘or’).  Note that the symbols “” and “” may also be without the equality bar.

  1. 1.

    |a|b  a-bab

  2. 2.

    |a|b  -bab

  3. 3.


These rules are valid for all real values of a and b.  For example, if one has a case


corresponding the rule 2, this inequality seems to be impossible since no absolute value is negative; but now also the result  -(-5)<x<-5  given by the rule 2 is impossible — no real number is simultaneously greater than +5 and less than -5.

Examples.  We solve some inequalities with absolute values.

a)  |2x+1|>5x
2x+1<-5x  or  2x+1>5x   (rule 1)
7x<-1  or  -3x>-1
x<-1/7  or  x<1/3
x<1/3   (combined)

b)  8|x|+|x-2|>6
8x<-6+|x-2|  or  8x>6-|x-2|   (rule 1)
|x-2|>8x+6  or  |x-2|>6-8x
x-2<-8x-6  or  x-2>8x+6  or  x-2<-6+8x  or  x-2>6-8x  (rule 1 twice)
9x<-4  or  -7x>8  or  -7x<-4  or  9x>8
x<-4/9  or  x<-8/7  or  x>4/7  or  x>8/9
x<-4/9  or  x>4/7   (from the number line)

c)  |1-5x|3
-31-5x3   (rule 2)
-4-5x2   (subtracted 1 from all parts)
4/5x-2/5   (divided by -5)
-2/5x4/5   (rewritten from end to begin)

Title inequality with absolute values
Canonical name InequalityWithAbsoluteValues
Date of creation 2013-03-22 16:57:20
Last modified on 2013-03-22 16:57:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 97D40
Related topic AbsoluteValue
Related topic AbsoluteValueInequalities
Related topic OrderOfSixMeans