Solve the inequality on the number line? How would I solve the following inequality.
$x^2+10x \gt-24$
How would I solve it and put it in a number line?

$x^2+10x>−24$
iff $x^2-10x+25 > 1$
iff $(x-5)^2 > 1$.
(Note: "iff" means "if and only if".)
Because of the magical property of $1$ being its own square root
(funny how this happens, eh?)
this is true iff $|x-5| > 1$
which is true
iff $x<4$ or $x > 6$.
By completing the square like this,
you are implicitly solving the equation.
If you start with the inequality
$x^2-2bx > c$
($b=5$ and $c=-24$ in your case)
this becomes
$(x-b)^2 > c + b^2$.
If $c + b^2 < 0$,
all $x$ satisfy this;
otherwise this can be rewritten as
$|x-b| > \sqrt{c+b^2}$
for which the solutions are
$x < b-\sqrt{c+b^2}$
and
$x > b+\sqrt{c+b^2}$
If you start with the inequality
$x^2-2bx < c$
this becomes
$(x-b)^2 < c + b^2$.
If $c + b^2 < 0$,
no $x$ satisfies this;
otherwise this can be rewritten as
$|x-b| < \sqrt{c+b^2}$
for which the solutions are
$b-\sqrt{c+b^2}
 < x < b+\sqrt{c+b^2}$
