Problem 51 - Precalculus

Question

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x|>1$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1) \cup (1, \infty)\) on a number line.

Step 1: Understanding the Absolute Value Inequality

We are given the inequality \(|x| > 1\). The absolute value \(|x|\) represents the distance of \(x\) from 0 on the number line. The inequality \(|x| > 1\) means we are looking for all points on the number line that are more than 1 unit away from 0.

Step 2: Breaking Down the Absolute Value Inequality

The inequality \(|x| > 1\) translates to two separate inequalities because \(x\) can be greater than 1 or less than -1. Thus, the inequality can be rewritten as \(x > 1\) or \(x < -1\).

Step 3: Solving the Inequalities

Solve the inequalities separately. The first inequality \(x > 1\) implies that \(x\) can be any number greater than 1. The second inequality \(x < -1\) implies that \(x\) can be any number less than -1.

Step 4: Writing the Solution in Interval Notation

The solution to the inequality \(|x| > 1\) is all numbers greater than 1 and all numbers less than -1. Therefore, in interval notation, the solution can be expressed as \((-\infty, -1) \cup (1, \infty)\).

Step 5: Visualizing on the Number Line

On the number line, \((-\infty, -1)\) is represented as an interval that starts from negative infinity and goes up to -1, without including -1 (open circle at -1). Similarly, \((1, \infty)\) starts just after 1 and extends to positive infinity, without including 1 (open circle at 1).

Key Concepts

Absolute Value

The concept of absolute value is crucial in understanding how distances work in mathematics. Absolute value denotes the distance of a number from zero on the number line, regardless of direction. For any real number \(x\), the absolute value is represented as \(|x|\). It is expressed as:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

In simpler terms, if \(x\) is positive, its absolute value is \(x\). If \(x\) is negative, its absolute value is its positive counterpart. So, the absolute value turns a negative number into a positive one, reflecting its 'absolute' distance from zero.

For example, the absolute value of -5 is 5, as it is 5 units away from zero on the number line.

Interval Notation

Interval notation is a handy way to describe a set of numbers along the number line. It uses brackets and parentheses to specify which numbers belong to the set.

• Use open parentheses \((\) or \()\) to indicate that endpoints are not included (open interval).
• Use closed brackets \([\) or \(]\) to mean that endpoints are included (closed interval).

For instance, the inequality \(|x| > 1\) results in intervals that include neither -1 nor 1 but every number beyond those limits. Therefore, it's written as \((-\infty, -1) \cup (1, \infty)\).
This expression reads as a union of two intervals: all real numbers starting from negative infinity up to, but not including -1, and continuing from just beyond 1 to positive infinity.

Number Line

The number line is a visual representation of real numbers aligned in order. It extends infinitely in both directions, with zero typically placed at the center. When discussing inequalities like \(|x| > 1\), the number line helps us visualize solutions:

• The numbers less than -1 are shown stretching leftward from -1.
• Numbers greater than 1 extend rightward from 1.

Open circles often indicate numbers that are not part of the solution set. For this example, -1 and 1 have open circles on the number line, demonstrating that they are not included in the solution for \(|x| > 1\), while extending endlessly towards negative and positive infinity.

Real Numbers

Real numbers encompass all the numbers we interact with daily, which include both rational numbers (fractions, integers) and irrational numbers (such as square roots of non-perfect squares).
These numbers can be plotted on the number line. The idea behind solving inequalities like \(|x| > 1\) is finding all real numbers meeting the inequality conditions. Therefore, the solution involves identifying these real number points and representing them in interval notation. Overall, real numbers cover an infinite range including negative numbers, zero, and positive numbers, providing a complete field for solving various mathematical problems.