squaring condition for square root inequality

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

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

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

Example. Solve the inequality

\displaystyle\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:

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

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

2x+3\;>\;x^{2}

x^{2}-2x-3\;<\;0

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 $-1<x<3$ (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.

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