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Chapter 2: Problem 52

Solve each rational inequality 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
So the solution to the inequality \(\frac{x+4}{x} > 0\), in interval notation is \((-鈭�, -4) 鈭� (0, 鈭�)\).

Step by step solution

01

Define the inequality

First, write down the given inequality \(\frac{x+4}{x}>0\). Keep in mind that the denominator of the fraction can not be 0, meaning we must exclude \(x=0\) from the possible solutions.
02

Determine the critical numbers

The critical numbers are the solutions of the equation \(\frac{x+4}{x}=0\) and \(x = 0\) (the value that make the denominator zero). Solving the first equation gives us \(x = -4\). So the critical numbers are \(x = 0\) and \(x = -4\).
03

Test the intervals

Now, need to test the sign of the function in the intervals \((-鈭�, -4)\), \((-4, 0)\), and \((0, 鈭�)\). By choosing a representative number from each interval, for instance, let's choose -5, -1 and 1, we can test the sign of the function: \(\frac{x+4}{x}\) > 0. At \(x = -5\) the result is positive, at \(x = -1\) the result is negative and at \(x = 1\) the result returns to be positive. Hence, in interval notation the solution set is \((-鈭�, -4) 鈭� (0, 鈭�)\).
04

Graphing the solution

Draw a number line and mark the critical numbers -4 and 0. Highlight the intervals \((-鈭�, -4)\) and \((0, 鈭�)\) on the number line because these are the values of \(x\) that makes \(\frac{x+4}{x} > 0\). The plot will show a filled circle at \(x = -4\) and \(x = 0\), and rays extending from these points to -鈭� and 鈭�.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

Find the inverse of \(f(x)=x^{3}+2\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

What is a rational inequality?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of this direct variation equation that has a positive constant of variation shows one variable increasing as the other variable decreases.
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