Problem 19 - Thomas Calculus

Question

Solve the inequalities in Exercises \(19-34,\) expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line. $$ |x|<2 $$

Step-by-Step Solution

Answer

The solution is \((-2, 2)\).

Step 1: Understanding the Inequality

The inequality given is \(|x| < 2\). The inequality \(|x| < 2\) implies that the distance of \(x\) from 0 is less than 2. This can be rewritten as two separate linear inequalities: \(-2 < x < 2\). This represents the numbers between -2 and 2.

Step 2: Expressing the Solution Set

The solution to \(-2 < x < 2\) can be expressed in interval notation. It includes all the numbers between -2 and 2, not including -2 and 2 themselves. Therefore, the interval notation for this solution is \((-2, 2)\).

Step 3: Graphing the Solution Set

On a number line, the interval \((-2, 2)\) is graphed as an open interval between -2 and 2. Draw a number line and shade the region between -2 and 2 without including -2 and 2 themselves. The points -2 and 2 are indicated with open circles to show they are not included in the solution set.

Key Concepts

Understanding Absolute Value

The idea of absolute value can be thought of as measuring the distance of a number from zero on a number line. This means when we talk about the absolute value of a number, we are considering how far away it is from zero, without being concerned about direction. The absolute value is always a non-negative number. For example,

• The absolute value of 3 is 3, written as \(|3| = 3\).
• The absolute value of -3 is also 3, written as \(|-3| = 3\).

In solving the inequality \(|x| < 2\), we are looking at all numbers whose distance from zero is less than 2. This concept forms the basis for transforming absolute value inequalities into separate linear inequalities which are often easier to solve.

Expressing with Interval Notation

Interval notation is a convenient way of expressing the set of solutions for inequalities, especially ones involving ranges of numbers. For instance, the inequality \(-2 < x < 2\) describes numbers greater than -2 but less than 2. In interval notation, this is written as \((-2, 2)\).

• Parentheses \(()\) are used to indicate that the endpoints themselves are not included in the interval.
• Brackets \[[]\] are used if the endpoints are included, but here we only use parentheses since neither -2 nor 2 is part of the solution.

Interval notation helps in quickly writing and identifying the range of numbers in a solution set without having to draw them out on a line.

Visualizing with a Number Line

A number line offers a clear visual representation of an inequality's solution set. When dealing with an inequality like \(-2 < x < 2\), the open interval \((-2, 2)\) can be depicted on a number line that highlights values between -2 and 2. To graph this:

• First, draw a horizontal line and mark points for -2 and 2.
• Place open circles at these points to signify that they are not included in the solution set.
• Shade or draw a line segment between these points to show all numbers in between are included.

This method lets you see the span and limitations of a solution in a visually intuitive way.

Breaking Down Linear Inequalities

Linear inequalities, such as \(-2 < x < 2\), are mathematical statements that compare linear expressions. In this case, the absolute value inequality \(|x| < 2\) is converted into two linear inequalities, \(-2 < x\) and \(x < 2\). Here's how it breaks down:

• "Less than" translates into a range of values (a segment on the number line).
• Solving involves handling each part of the inequality separately, then combining them.

This transformation step simplifies solving, moving from an absolute context to a more straightforward linear one, where these inequalities can be understood and visualized on a number line or expressed mathematically with interval notation.