Solve the inequality $-4 > (x-5)(x+5) - (x-3)^2$ for $x$.

1. Problem Description

Solve the inequality $-4 > (x-5)(x+5) - (x-3)^2$ for $x$.

2. Solution Steps

First, expand the terms on the right side of the inequality.

$(x-5)(x+5) = x^2 - 25$

$(x-3)^2 = x^2 - 6x + 9$

Now substitute these expanded terms into the inequality:

$-4 > (x^2 - 25) - (x^2 - 6x + 9)$

Distribute the negative sign:

$-4 > x^2 - 25 - x^2 + 6x - 9$

Combine like terms:

$-4 > 6x - 34$

Add 34 to both sides of the inequality:

$-4 + 34 > 6x$

$30 > 6x$

Divide both sides by 6:

$\frac{30}{6} > x$

$5 > x$

This can also be written as $x < 5$.

3. Final Answer

$x < 5$