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$x^2-36>0$

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Solve advanced linear inequalities

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Q.

x^2-36>0

Factor Quadratic Expression: Factor the quadratic expression. $x^2 - 36 = (x - 6)(x + 6)$
Set Inequality: Set the inequality with the factored form. $\newline$ $(x - 6)(x + 6) > 0$
Determine Critical Points: Determine the critical points where the expression equals zero. $\newline$ $x - 6 = 0 \rightarrow x = 6$ $\newline$ $x + 6 = 0 \rightarrow x = -6$
Analyze Sign of Product: Analyze the sign of the product in each interval divided by the critical points: $(-\infty, -6)$ , $(-6, 6)$ , and $(6, \infty)$ .
Test $x = -7$ in $(-\infty, -6)$ : $(-7 - 6)(-7 + 6) = (-13)(-1) = 13 > 0$
Test $x = 0$ in $(-6, 6)$ : $(0 - 6)(0 + 6) = (-6)(6) = -36 < 0$
Test $x = 7$ in $(6, \infty)$ : $(-6, 6)$ $1$
Write Solution: Write the solution based on the sign analysis. $\newline$ The expression $(x - 6)(x + 6) > 0$ when $x < -6$ or $x > 6$ .

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