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Chapter 3: Problem 65

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

Short Answer

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

Step by step solution

01

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 \(x^2 + 2x - 36 < -12\).
02

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.
03

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 \( (6, \infty) \).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has vertical asymptotes given by x=-2 and x=2, a horizontal asymptote y=2, y -intercept at \frac{9}{2}, x -intercepts at -3 and 3, and y -axis symmetry.

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3} $$

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.
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