squaring condition for square root inequality

Of the inequalitiesMathworldPlanetmathab,

  • both are undefined when  a<0;

  • both can be sidewise squared when  a0  and  b0;

  • a>b  is identically true if  a0  and  b<0.

  • a<b  is identically untrue if  b<0;

The above theorem may be utilised for solving inequalities involving square roots.

Example.  Solve the inequality

2x+3>x. (1)

The reality condition  2x+30  requires that  x-112.  For using the theorem, we distinguish two cases according to the sign of the right hand side:

1:  -112x<0.  The inequality is identically true; we have for (1) the partial solution  -112x<0.

2:  x0.  Now we can square both , obtaining

x2-2x-3< 0

The zeros of x2-2x-3 are  x=1±2,  i.e. -1 and 3. Since the graph of the polynomial function is a parabola opening upwards, the polynomialPlanetmathPlanetmath attains its negative values when  -1<x<3 (see quadratic inequality).  Thus we obtain for (1) the partial solution  0x<3.

Combining both partial solutions we obtain the total solution

-112x< 3.
Title squaring condition for square root inequality
Canonical name SquaringConditionForSquareRootInequality
Date of creation 2013-03-22 17:55:56
Last modified on 2013-03-22 17:55:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 26D05
Classification msc 26A09
Synonym squaring condition
Related topic StrangeRoot