Finding Values Where x² Exceeds 16: Inequality Solution

The objective is to find the values of $x$ such that the inequality $x^2 - 16 > 0$ is satisfied.

Step 1: Factor the inequality expression.

The expression $x^2 - 16$ can be factored using the difference of squares formula:

$x^2 - 16 = (x - 4)(x + 4)$.

Step 2: Determine the critical points.

Set the factors equal to zero to find the critical points:

• $x - 4 = 0$ gives $x = 4$.
• $x + 4 = 0$ gives $x = -4$.

Step 3: Analyze the sign changes on the number line.

We test the intervals defined by the critical points $-4$ and $4$ on a number line: $(-∞, -4)$, $(-4, 4)$, $(4, ∞)$.

Choose a test point from each interval and substitute into the factored expression to check the sign.

• For $x = -5$ (interval $(-∞, -4)$): $(x - 4)(x + 4) = (-5 - 4)(-5 + 4) = (-9)(-1) > 0$.
• For $x = 0$ (interval $(-4, 4)$): $(x - 4)(x + 4) = (0 - 4)(0 + 4) = (-4)(4) < 0$.
• For $x = 5$ (interval $(4, ∞)$): $(x - 4)(x + 4) = (5 - 4)(5 + 4) = (1)(9) > 0$.

Step 4: Extract the solution.

The inequality $x^2 - 16 > 0$ holds true in the intervals where the product is positive, which are $(-∞, -4) \cup (4, ∞)$.

Therefore, the solution to the inequality is $x < -4$ or $x > 4$.

The correct choice is $x < -4, 4 < x$.