squaring condition for square root inequality
Of the inequalities^{} $\sqrt{a}\lessgtr b$,

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both are undefined when $$;

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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 $$.

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$$ is identically untrue if $$;
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 $x\geqq 1\u2064\frac{1}{2}$. 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
$$2x+3>{x}^{2}$$ 
$$ 
The zeros of ${x}^{2}2x3$ are $x=1\pm 2$, i.e. $1$ and $3$. Since the graph of the polynomial function is a parabola opening upwards, the polynomial^{} attains its negative values when $$ (see quadratic inequality). Thus we obtain for (1) the partial solution $$.
Combining both partial solutions we obtain the total solution
$$ 
Title  squaring condition for square root inequality 

Canonical name  SquaringConditionForSquareRootInequality 
Date of creation  20130322 17:55:56 
Last modified on  20130322 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 