PlanetMath: squaring condition for square root inequality

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squaring condition for square root inequality

(Theorem)

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 .
$\sqrt{a} < b$ is identically untrue if ;

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:

$1^{\underline{o}}$ : $-1\frac{1}{2} \leqq x < 0$ . 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

$\displaystyle 2x+3 > x^2$

$\displaystyle x^2-2x-3 < 0$

The zeros of $x^2\!-\!2x\!-\!3$ are $x = 1\pm2$ , 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 $0 \leqq x < 3$ .

Combining both partial solutions we obtain the total solution

$\displaystyle -1\frac{1}{2} \leqq x < 3.$

"squaring condition for square root inequality" is owned by pahio.

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See Also: strange root

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squaring condition

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Cross-references: quadratic inequality, negative, polynomial, parabola, polynomial function, graph, square, solution, right hand side, square roots, inequalities

This is version 2 of squaring condition for square root inequality, born on 2008-03-21, modified 2008-03-21.
Object id is 10427, canonical name is SquaringConditionForSquareRootInequality.
Accessed 145 times total.

Classification:

AMS MSC:	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	26D05 (Real functions :: Inequalities :: Inequalities for trigonometric functions and polynomials)

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