Problem 20 - 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. $$ |x| \leq 2 $$

Step-by-Step Solution

Answer

The solution is \([-2, 2]\), all numbers between -2 and 2 inclusive.

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|x| \leq 2\). This indicates that the distance of a number \(x\) from zero on the number line is less than or equal to 2.

Step 2: Break Down the Absolute Value Inequality

An inequality of the form \(|x| \leq a\) is equivalent to the compound inequality \(-a \leq x \leq a\). So for \(|x| \leq 2\), we can write: \(-2 \leq x \leq 2\).

Step 3: Express the Solution in Interval Notation

The inequality \(-2 \leq x \leq 2\) represents all real numbers \(x\) between -2 and 2, inclusive of both endpoints. In interval notation, this is expressed as \([-2, 2]\).

Step 4: Visualize the Solution on a Number Line

On a number line, represent the interval \([-2, 2]\) by marking solid dots at -2 and 2 and shading the line segment between them, as it includes all numbers from -2 to 2.

Key Concepts

Interval Notation

Interval notation is a mathematical way of expressing subsets of real numbers. When dealing with inequalities, it's an efficient method to compactly denote the range of solutions. For absolute value inequalities, which often result in a range of values, interval notation makes it easy to express the start and end of a set of numbers.

The inequality \(-2 \leq x \leq 2\) means that \(x\), the variable in the inequality, can take any value from -2 to 2, including -2 and 2 themselves. Using interval notation, we express this range by writing \([-2, 2]\). The brackets \([ ]\) signify that the endpoints -2 and 2 are included in the solution set, known as 'inclusive' endpoints. Whenever the bounds are not included, parentheses \(( )\) would be used instead, indicating an 'exclusive' endpoint.

• Inclusive endpoints: Brackets \([ ]\) are used.
• Exclusive endpoints: Parentheses \(( )\) are used.
• Mixed endpoints: A combination of brackets and parentheses, such as \((2, 5]\).

Interval notation is crucial in both expressing and understanding the breadth of solutions an inequality can have.

Number Line Representation

Visualizing solutions on a number line can greatly aid in understanding the range of values an inequality encompasses. For the inequality \(|x| \leq 2\), which translates to \(-2 \leq x \leq 2\), a number line effectively shows this range.

To represent \([-2, 2]\) on a number line:

• Draw a horizontal line with numbers marked in a sequence that includes -2 and 2.
• Place a solid dot on -2 and another on 2. These solid dots highlight that -2 and 2 are part of the solution set.
• Shade or color the segment of the line between -2 and 2 to indicate all numbers in this interval are solutions.

This representation makes it easy to see at a glance which numbers satisfy the inequality, providing a clear and immediate understanding of the solution set.

Compound Inequalities

Compound inequalities involve multiple inequalities being satisfied at the same time. In the case of absolute value inequalities such as \(|x| \leq 2\), breaking it down into a compound inequality helps simplify it into two straightforward pieces, \-2 \leq x \leq 2\.

Compound inequalities can be written in different forms depending on the scenario, notably:

• Conjunction: Both conditions must be true simultaneously, often using "and." For example, \(-2 \leq x \text{ and } x \leq 2\).
• Disjunction: At least one of the conditions must be true, using "or." An example is \(x < -2 \text{ or } x > 2\).

By converting an absolute value inequality to a compound inequality, you simplify the range of possible values into an easy-to-understand expression. This method is particularly helpful when visualizing the ranges on a number line or expressing them in interval notation. Understanding this concept lays the foundation for solving more complex inequalities.