Solving quadratic inequality x2&gt;0

Step 1We have thatfor x=0⇒x2=0for x≠0⇒x2&gt;0and the proof is complete by exhaustion.Following your idea, using that x2=|x|, we can take the square root both sides to obtainx2&gt;0⇔x2&gt;0⇔|x|&gt;0which is always true for x≠0.

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High School
Algebra
Algebra I
Quadratic function and equation

dayncNonow04r

Answered question

2022-04-03

Solving quadratic inequality

x^{2} > 0

Answer & Explanation

Janessa Foster

Beginner2022-04-04Added 12 answers

Step 1
Caution,
$a^{2} > b$
does not imply
$a > \pm \sqrt{b}$
But
$\pm a > \sqrt{b}$
is correct. (With a somewhat sloppy notation.)
Step 2
More rigorously
$a^{2} > b$
$\Leftrightarrow (a - \sqrt{b}) (a + \sqrt{b}) > 0$
$\Leftrightarrow (a > \sqrt{b} \land a > - \sqrt{b}) \lor (a < \sqrt{b} \land a < - \sqrt{b})$
$\Leftrightarrow a > \sqrt{b} \lor a < - \sqrt{b} .$

Jesse Gates

Beginner2022-04-05Added 19 answers

Step 1
We have that
for

x = 0 \Rightarrow x^{2} = 0

for

x \neq 0 \Rightarrow x^{2} > 0

and the proof is complete by exhaustion.
Following your idea, using that

\sqrt{x^{2}} = | x |

, we can take the square root both sides to obtain

x^{2} > 0 \Leftrightarrow \sqrt{x^{2}} > \sqrt{0} \Leftrightarrow | x | > 0

which is always true for

x \neq 0

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