Problem 65 - Algebra and Trigonometry

Question

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \left|x^{2}+2 x-36\right|>12 $$

Step-by-Step Solution

Answer

The solutions to the inequality are \( x < -8\) and \( x > 6\).

Step 1: Remove the Absolute Value

Considering the nature of absolute value, the equation will be broken down into two separate equations. These will be: \(x^2 + 2x - 36 > 12\) and also \(x^2 + 2x - 36 < -12\).

Step 2: Solve Each Inequality

First solve \(x^2 + 2x - 36 > 12\). This simplifies to \(x^2 + 2x - 48 > 0\). Factoring the quadratic equation, \( (x - 6)(x + 8) > 0\). Setting the factors equal to zero gives two critical points, \( x = 6\) and \( x = -8\) which divide the number line into three regions. Test a point in each area to see where the inequality is satisfied. The second inequality \(x^2 + 2x - 36 < -12\) has no solution because a squared term is always positive and cannot be less than a negative value.

Step 3: Graph the Solution on a Real Number Line

Graphing the solutions -8 and 6 from the first inequality on a number line, solutions exist in the regions where the first inequality is true, corresponding to the intervals \( (-\infty, -8) \) and \( (6, \infty) \).