Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ |x| \geq 5 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -5] \cup [5, \infty)\).

Step 1: Understanding the Absolute Value Inequality

The inequality \(|x| \geq 5\) means that the absolute value of \(x\) is at least 5. Absolute value inequalities like this one are typically broken into two separate inequalities.

Step 2: Breaking Down the Inequality

The inequality \(|x| \geq 5\) translates to two possibilities: 1. \(x \geq 5\)2. \(x \leq -5\)

Step 3: Solving Each Component

Solve the first part \(x \geq 5\). This means any \(x\) value greater than or equal to 5 satisfies the inequality. Solve the second part \(x \leq -5\). This means any \(x\) value less than or equal to -5 satisfies the inequality.

Step 4: Writing the Solution in Interval Notation

Using interval notation, combine both parts of the inequality:1. For \(x \geq 5\), the interval is \([5, \infty)\).2. For \(x \leq -5\), the interval is \((-\infty, -5]\).Therefore, the solutions can be expressed in interval notation as \((-\infty, -5] \cup [5, \infty)\).

Key Concepts

Interval Notation

Interval notation is a shorthand method of representing a range of numbers, often used for expressing solutions to inequalities. It provides a clear and concise way to describe a set of numbers without listing them individually. In mathematics, interval notation uses brackets and parentheses to show closed and open intervals, respectively. Here's how:

• Square brackets, like \([a, b]\), indicate that the endpoints \(a\) and \(b\) are included in the set (closed interval).
• Round parentheses, as in \((a, b)\), mean that the endpoints are not included in the set (open interval).

For example, the solution to \(|x| \geq 5\) consists of two separate intervals written in interval notation. For the inequality segment \(x \geq 5\), the solution is \([5, \infty)\), indicating that \(x\) can take any value from 5 to infinity, including 5 itself. For \(x \leq -5\), the solution \((-\infty, -5]\) includes all values less than or equal to -5.
When combined, these intervals, \((-\infty, -5] \cup [5, \infty)\), fully represent all values of \(x\) that satisfy the initial inequality.

Inequality Solutions

Inequality solutions involve finding a range of values that satisfy a particular inequality. When dealing with an absolute value inequality like \(|x| \geq 5\), we must consider both the positive and negative scenarios.
This is because absolute values represent the distance of a number from zero on the number line, making it inherently non-negative.

To solve \(|x| \geq 5\), it is necessary to break the problem into two separate inequalities due to the nature of absolute values:

• First, solve for \(x\) when the positive aspect is considered: \(x \geq 5\).
• Second, solve for \(x\) when the negative aspect is considered: \(x \leq -5\).

This process ensures that all possible values of \(x\) that make the original inequality true are accounted for. Each solution represents a different segment on the number line, which together form the complete solution when combined in interval notation.

Mathematical Inequalities

Mathematical inequalities are statements about the relative size or order of two values. Inequalities use symbols like \(\geq\), \(\leq\), \(>\), and \(<\) to denote these relationships.
In terms of absolute value inequalities, such as \(|x| \geq 5\), they provide a way to describe ranges rather than specific numbers. The absolute value, denoted by vertical bars \(|\cdot|\), gives only the magnitude of a number, effectively ignoring its sign.

As such, when you solve inequalities involving absolute values:

• You must consider both parts of the inequality: one where the absolute value expression is positive or zero, and one where it is negative.
• This ensures that both the possible positive and possible negative values are included in the solution.
• The entire process highlights the comprehensive nature of inequalities, allowing for a broader range of numbers than equations.

Understanding these concepts makes inequalities a powerful tool in mathematics for describing situations where ranges of values are acceptable solutions, as demonstrated with the inequality \(|x| \geq 5\).