Problem 38 - Thomas Calculus

Question

Solve the inequalities in Exercises \(35-42 .\) Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result \(\sqrt{a^{2}}=|a|\) as appropriate. $$ \frac{1}{9} < x^{2} < \frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \( \left( \frac{1}{3}, \frac{1}{2} \right) \cup \left( -\frac{1}{2}, -\frac{1}{3} \right) \).

Step 1: Understand the Inequality

We are given a compound inequality \( \frac{1}{9} < x^2 < \frac{1}{4} \). This means that the square of \( x \) is between \( \frac{1}{9} \) and \( \frac{1}{4} \). Our task is to find the values of \( x \) that make this inequality true.

Step 2: Solve the Left Inequality

Let's first solve the inequality \( x^2 > \frac{1}{9} \). Rewrite this inequality as \( x^2 - \frac{1}{9} > 0 \). Thus, \( x^2 > \frac{1}{9} \) implies \( x > \frac{1}{3} \) or \( x < -\frac{1}{3} \).

Step 3: Solve the Right Inequality

Now solve \( x^2 < \frac{1}{4} \). Rewrite this as \( x^2 - \frac{1}{4} < 0 \). Thus, \( x^2 < \frac{1}{4} \) implies \( -\frac{1}{2} < x < \frac{1}{2} \).

Step 4: Combine the Solutions

The solution to the original compound inequality \( \frac{1}{9} < x^2 < \frac{1}{4} \) is the intersection of the solution sets from the left and right inequalities. Therefore, the solution in terms of \( x \) will be the values that satisfy both: \( \frac{1}{3} < x < \frac{1}{2} \) or \(-\frac{1}{2} < x < -\frac{1}{3} \).

Step 5: Express as Intervals

Express the solution as two intervals: \( \left( \frac{1}{3}, \frac{1}{2} \right) \cup \left( -\frac{1}{2}, -\frac{1}{3} \right) \).

Step 6: Show on the Number Line

On a number line, mark the intervals \( \left( \frac{1}{3}, \frac{1}{2} \right) \) and \( \left( -\frac{1}{2}, -\frac{1}{3} \right) \) as open intervals where \( x \) is valid. These are shown as two separate segments without endpoints.

Key Concepts

Compound Inequality

A compound inequality involves two inequalities that are combined in one statement. Here, we have \( \frac{1}{9} < x^2 < \frac{1}{4} \).This means that the square of \( x \) must be greater than \( \frac{1}{9} \) as well as less than \( \frac{1}{4} \). Essentially, you are looking for the values of \( x \) that satisfy both conditions part of the same expression.

To solve a compound inequality, you can break it down into two separate inequalities, solve them individually, and then find the intersection of their solution sets.

• The first part is \( x^2 > \frac{1}{9} \), which simplifies to \( x > \frac{1}{3} \) or \( x < -\frac{1}{3} \).
• The second part is \( x^2 < \frac{1}{4} \), giving us \( -\frac{1}{2} < x < \frac{1}{2} \).

The final solution is the overlap or intersection of these two ranges.

Interval Notation

Interval notation is a convenient way of writing the solution to an inequality as a range of numbers. It involves using parentheses \(()\) and brackets \([]\) to express open and closed intervals.

For our compound inequality, the solution set is expressed as two open intervals:

• \((\frac{1}{3}, \frac{1}{2})\) indicates numbers greater than \(\frac{1}{3}\) and less than \(\frac{1}{2}\).
• \((-\frac{1}{2}, -\frac{1}{3})\) indicates numbers less than \(-\frac{1}{3}\) and greater than \(-\frac{1}{2}\).

These intervals are joined by a union symbol \((\cup)\), showing that numbers within either range satisfy the original inequality. This is one great way to compactly represent all potential solutions on a numerical line.

Number Line Representation

Number line representation is a visual way to show solutions to inequalities. This can help you quickly and easily see the range of solutions.

On a number line:

• The part \((\frac{1}{3}, \frac{1}{2})\) is represented as a segment between \(\frac{1}{3}\) and \(\frac{1}{2}\), with open circles on \(\frac{1}{3}\) and \(\frac{1}{2}\) to show that these endpoints are not included.
• Similarly, \((-\frac{1}{2}, -\frac{1}{3})\) is shown with a segment between \(-\frac{1}{2}\) and \(-\frac{1}{3}\), also with open circles.

This graphical illustration gives a quick and easy reference to see where \( x \) can be found on the number line, reflecting the solution logic presented in interval notation.

Square Root Property

The square root property is a crucial concept when dealing with inequalities involving square terms, such as \( x^2 \).

This property indicates that \( \sqrt{a^2} = |a| \). Thus, when we are solving inequalities like \( x^2 > \frac{1}{9} \) or \( x^2 < \frac{1}{4} \), we need to consider both positive and negative roots.

• For \( x^2 > \frac{1}{9} \), the solutions translate to \( x > \frac{1}{3} \) and \( x < -\frac{1}{3} \), indicating two potential ranges.
• For \( x^2 < \frac{1}{4} \), the result yields \( -\frac{1}{2} < x < \frac{1}{2} \) due to the square root covering both potential sign scenarios.

Understanding how to correctly apply the square root property ensures you capture all possible solutions, including those on the negative number spectrum.