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Chapter 3: Problem 52

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{x}>0 $$

Short Answer

Expert verified
The solution set for the inequality \( \frac{x+4}{x} > 0 \) in interval notation is \( (-\infty, -4) \cup (0, +\infty) \).

Step by step solution

01

Determine the Critical Values

Critical values are those values that make the rational inequality equal to zero or undefined. In this case, the critical values for the inequality \( \frac{x+4}{x}> 0 \) are -4 (where the numerator equals to zero) and 0 (where the denominator equals to zero).
02

Test the Critical and Non-Critical Intervals

The critical values split the number line into intervals: \(-\infty , -4\), \(-4, 0\), and \(0, +\infty\). Choose representatives in each one of these intervals and substitute back into the inequality to check where it holds. As an example, let's choose -5 for the interval \(-\infty , -4\), -1 for \(-4, 0\), and 1 for \(0, +\infty\). After testing these values, we see that the inequality holds for intervals \(-\infty , -4\) and \(0, +\infty\).
03

Express the solution in Interval Notation and Graph

Now that we have determined the solution set of the inequality, it can be expressed in interval notation as \((-\infty, -4) \cup (0, +\infty)\). This is also the range of solutions to our initial problem. Graphing this solution would involve graphing these intervals on a real number line, showing an open circle at -4 and 0 and shading to the left of -4 and to the right of 0.

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Most popular questions from this chapter

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 a vertical asymptote given by x=3, a horizontal asymptote y=0, y -intercept at -1, and no x -intercept.

Write an equation in point-slope form and slope-intercept form of the line passing through \((-10,3)\) and \((-2,-5)\) (Section \(2.3,\) Example 3 )

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

What is a rational function?

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. $$ \frac{x}{2 x+6}-\frac{9}{x^{2}-9} $$
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