Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(\frac{1}{x}>x\)

Step-by-Step Solution

Answer

The solution to the inequality \(\frac{1}{x}>x\) is \(x \in \(-\infty, -1)\) union \((0, 1)\). The plot will resemble segments on the real number line at \(-\infty\) to \(-1\) and from \(0\) to \(1\).

Step 1: Simplify the inequality

Rewrite the inequality in standard form (zero on one side) by subtracting \(x\) from both sides: \(\frac{1}{x} - x > 0\) or equivalently \(\frac{1}{x} - x = 0\).

Step 2: Find the common denominator

To move forward, it's easier to deal with the inequality when the fractions are removed. This could be achieved by finding a common denominator, which is \(x\) in this case: \(\frac{1-x^2}{x} > 0\).

Step 3: Find critical values

To find the critical values, set the equation \(\frac{1-x^2}{x} = 0\) and solve. The solutions are the values of \(x\) which make one part of the equation equal to zero. So \(x^2 = 1\) or \(x = 0\). Solving for \(x\) leads to \(x = -1,0,+1\).

Step 4: Test the intervals

Following from Step 3, the critical values segment the real number line into four intervals: \(-\infty,-1\), \(-1,0\), \(0,1\) and \(1,\infty\). Pick a test point from each interval and substitute it into the inequality \(\frac{1-x^2}{x}\). If the result is greater than zero, then that interval is part of the solution set.

Step 5: Plot the solution set

After determining the intervals that are part of solution set in Step 4, plot the solution set on a real number line. The plot should include the intervals in which the inequality is satisfied, and exclude the critical points where the inequality is not defined.

Key Concepts

Critical Values in an Inequality

In solving inequalities like \( \frac{1}{x} > x \), we encounter critical values, which are key points that divide the number line into intervals. These points are determined by setting the expression equal to zero or identifying where it is undefined. In this exercise, the expression \( \frac{1-x^2}{x} \) becomes zero or is undefined when \( x = -1, 0, \) and \( 1 \).

- These critical values are essential in determining the behavior of the inequality.- At \( x = 0 \), the expression is undefined since division by zero is not possible. - At \( x = -1 \) and \( x = 1 \), the expression equals zero. Understanding critical values helps you identify which intervals to test in order to determine where the inequality holds true. They are the jumping-off points for checking the inequality across the range of potential solutions.

Understanding the Solution Set

The solution set of an inequality represents all the values that satisfy it. Once the critical values divide the number line into intervals, each of these intervals must be tested to check whether they satisfy the inequality. For example, testing a point from each interval determined by our critical values, such as \( \frac{1-x^2}{x} > 0 \), helps us figure out where the inequality holds.

- You must pick a test point from each interval, e.g., \(-2, -0.5, 0.5, 2\), and plug it into the inequality.- If the result is positive (greater than zero), that interval is part of the solution set.- Intervals that yield negative results are left out. The solution set of \( \frac{1}{x}>x \) includes the intervals where the inequality is true, showing the actual range of values as parts of the real number line where our inequality holds.

Using the Real Number Line

The real number line is a simple yet powerful tool in visualizing inequalities. When plotting solutions, it's a continuous line representing all real numbers. Using critical values, the line is split into segments that you test and then mark as inclusive or exclusive parts of your solution.

- Graphing highlights intervals segmenting the real number line determined by critical values.- Points corresponding to the critical values such as \( -1, 0, \) and \( 1 \) are represented but may or may not be part of the solution.- Clearly illustrate each interval's inclusion in the solution set, using open or closed circles to show where the critical points lie in relation to your solution.The real number line provides a visual aid that helps to better understand where solutions to inequalities exist and to ensure they are accurately expressed.