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

• both are undefined when $a < 0$ ;
• both can be sidewise squared when $a \geqq 0$ and $b \geqq 0$ ;
• $\sqrt{a} > b$ is identically true if $a \geqq 0$ and $b < 0$ .
• $\sqrt{a} < b$ is identically untrue if $b < 0$ ;

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

Example. Solve the inequality

(1)

$1^{\underline{o}}$ : $-1\frac{1}{2} \leqq x < 0$ . The inequality is identically true; we have for (1) the partial solution $-1\frac{1}{2} \leqq x < 0$ .

$2^{\underline{o}}$ : $x \geqq 0$ . Now we can square both sides, obtaining $$2x+3 > x^2$$ $$x^2-2x-3 < 0$$ The zeros of $x^2\!-\!2x\!-\!3$ are $x = 1\pm2$ , i.e. $-1$ and $3$ . Since the graph of the polynomial function is a parabola opening upwards, the polynomial attains its negative values when $-1 < x < 3$ (see quadratic inequality). Thus we obtain for (1) the partial solution $0 \leqq x < 3$ .

Combining both partial solutions we obtain the total solution $$-1\frac{1}{2} \leqq x < 3.$$