I am doing a problem and am not sure why the final step is the following
$x^{2}<4 <=> -2<x<2$
When simplifying
$x^{2}<4$ it becomes $x<±2$ which is the same as $x<2$, $x<-2$
but this does not match the actual inequality of $-2<x<2$, since $x>-2$ not $x<-2$ which is one of the inequalities
obtained in the line above
My question:
Inequalities are swapped when multiplying & dividing by negative numbers, so is the reason why $x^{2}<4 <=> -2<x<2$
because square rooting both sides of $x^{2} <4$ give $x<2$ AND $x >-2$ since $4$ became $-2$ so it is like the inequality rule of dividing by a negative number?
Hence the sign is flipped when considering $√(4)=-2$? $<=> 4/-2=-2$?
What I think:
$x^{2}<4$ gives four possible inequalities:
(1)$x<2$
(2)$-x<-2=>x>2$
(3)$-x<2=>x>-2$
(4)$x<-2$.
By inspection, (1) & (3) are the actual inequalities and (2) & (4) are false solutions.
(1) & (3) = $x<2$, $x>-2$ $<=>$ $-2<x<2$
Is this the reason why??
).
Best Answer
Care must be exercised when considering negative square roots. For example, we have $9\gt4$, but not $-3\gt-2$.
So we can only take positive square roots.
From $x^2\lt4$ and $(-x)^2\lt4$ this gives two equations: