squaring condition for square root inequality

Of the inequalities  $\sqrt{a}\lessgtr b$,

• both are undefined when  ;

• both can be sidewise squared when  $a\geqq 0$  and  $b\geqq 0$;

• $\sqrt{a}>b$  is identically true if  $a\geqq 0$  and  .

•   is identically untrue if  ;

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

Example.  Solve the inequality

$\sqrt{2x+3}>x.$(1)

The reality condition  $2x+3\geqq 0$  requires that  $x\geqq -1⁤\frac{1}{2}$.  For using the theorem, we distinguish two cases according to the sign of the right hand side:

$1^{\circ }$:  .  The inequality is identically true; we have for (1) the partial solution  .

$2^{\circ }$:  $x\geqq 0$.  Now we can square both , obtaining

$$2x+3>x^2$$

The zeros of $x^2-2x-3$ are  $x=1\pm 2$,  i.e. $-1$ and $3$. 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