Question

use a graphing calculator to solve the following inequality.
\\(\frac{1}{2}(x + 4)^2 - 7 > 43\\)

Explanation:

Step1: Isolate the squared term

Add 7 to both sides.
$\frac{1}{2}(x+4)^2 > 43 + 7$
$\frac{1}{2}(x+4)^2 > 50$

Step2: Eliminate the coefficient

Multiply both sides by 2.
$(x+4)^2 > 50 \times 2$
$(x+4)^2 > 100$

Step3: Take square roots

Solve for $x+4$ (consider both roots).
$x+4 > 10 \quad \text{or} \quad x+4 < -10$

Step4: Solve for x

Subtract 4 from all parts.
$x > 10 - 4 \quad \text{or} \quad x < -10 - 4$
$x > 6 \quad \text{or} \quad x < -14$

Answer:

$x < -14$ or $x > 6$ (in interval notation: $(-\infty, -14) \cup (6, \infty)$)