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Chapter 2: Problem 40

Solve the inequality. Then graph the solution set. $$\frac{x^{2}-1}{x}<0$$

Short Answer

Expert verified
The solution is \(-\infty < x < -1\) or \(0 < x < 1\).

Step by step solution

01

Express common factors

Express the numerator and denominator as products of their factors. Convert the quadratic \(x^{2}-1\) into difference of squares \(\left(x-1\right)\left(x+1\right)\) to get \(\frac{\left(x-1\right)\left(x+1\right)}{x}<0\)
02

Determine critical values

Set the numerator and the denominator to 0 and solve the equations to find the critical values. This leads to \(x = 1, -1, 0\). These critical values split the number line into four intervals: \(-\infty < x < -1\), \(-1 < x < 0\), \(0 < x < 1\), \(1 < x < \infty\).
03

Test points

Pick a value from each of the four intervals to test the sign of the inequality. For \(-\infty < x < -1\), let's test with \(x = -2\), and we get \((\(-2 - 1)(-2 + 1)/-2\) > 0. Thus in this interval, inequality holds true. Repeat the process for each interval.
04

Conclude solution

After testing all the possible intervals, we find that the inequality holds true for \(-\infty < x < -1\) and \(0 < x < 1\). However, remember that for inequalities, the critical points, where the denominator is 0, must be excluded. Hence, the final solution of the inequality is \(-\infty < x < -1\) or \(0 < x < 1\).

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