squaring condition for square root inequality
Of the inequalities ,
both are undefined when ;
both can be sidewise squared when and ;
is identically true if and .
is identically untrue if ;
The above theorem may be utilised for solving inequalities involving square roots.
Example. Solve the inequality
The reality condition requires that . For using the theorem, we distinguish two cases according to the sign of the right hand side:
: . The inequality is identically true; we have for (1) the partial solution .
: . Now we can square both , obtaining
The zeros of are , i.e. and . Since the graph of the polynomial function is a parabola opening upwards, the polynomial attains its negative values when (see quadratic inequality). Thus we obtain for (1) the partial solution .
Combining both partial solutions we obtain the total solution
|Title||squaring condition for square root inequality|
|Date of creation||2013-03-22 17:55:56|
|Last modified on||2013-03-22 17:55:56|
|Last modified by||pahio (2872)|