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Chapter 2: Problem 65

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

Short Answer

Expert verified
The solution to the inequality \(|x^2+2x-36|>12\) is \(x < -6\) or \(x > 4\).

Step by step solution

01

Create Two Separate Inequalities.

For any equation \(|x|> a\), we can create two separate inequalities: \(x> a\) and \(x< -a\). So, we can write the given inequality \(|x^2+2x-36|>12\) as two separate inequalities: \(x^2+2x-36 > 12\) and \(x^2+2x-36 < -12\)
02

Solve Each Inequality Separately.

Now, solve each inequality. First, solve \(x^2+2x-36 > 12\). To do this, subtract 12 from both sides to obtain \(x^2+2x-24 > 0\). Factoring gives \((x-4)(x+6) > 0\). Use test points in each interval to determine the signs: (-鈭�,-6), (-6,4), (4,鈭�). This gives a solution of \(x < -6\) or \(x > 4\). Next, solve \(x^2+2x-36 < -12\). Subtracting -12 from both sides gives \(x^2+2x-24 < 0\). Factoring results in \((x-4)(x+6) < 0\). Test points in each interval give the conclusion that no \(x\) satisfies this inequality.
03

Graph the Solution on a Real Number Line.

Now graph the solution set. Draw a number line, mark the points x=-6 and x=4. Shade the regions \(x < -6\) and \(x > 4\). The shaded area represents the solution.

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