Problem 98 - Introductory Algebra for College Students

Question

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\) The distance from 0 to \(x\) on a number line is greater than 3 .

Step-by-Step Solution

Answer

The inequality that describes the sentence 'The distance from 0 to \(x\) on a number line is greater than 3' is \(|x| > 3\), which represents all the locations on the number line where \(x\) is less than -3 or greater than 3.

Step 1: Understand the Statement

The statement 'The distance from 0 to \(x\) on a number line is greater than 3' implies that \(x\) is more than 3 units away from 0, either to the right (positive side) or to the left (negative side). This is essentially the definition of the absolute value: the distance of \(x\) to zero.

Step 2: Translate the Statement into an Inequality

Translate the given statement into mathematical form using the concept of absolute value. Here, '|x|' represents the distance of \(x\) from 0, and it is given that this distance is greater than 3. Hence, this can be written as \(|x| > 3\).

Step 3: Identifying the locations on the number line

Since the inequality is \(|x| > 3\), there are two possible regions where \(x\) might be located on a number line: one is to the right of 3 and the other to the left of -3. This is because the absolute value of any \(\{x: x > 3 \}\) or \(\{x: x < -3 \}\) will be greater than 3. The inequality thus describes all real numbers greater than 3 or less than -3.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to compare two values. They show that one value is larger or smaller than another. For example, if we know that \(|x| > 3\), it tells us that the absolute value of \(|x|\) is greater than 3.

There are some key symbols used in inequalities:

• \(< > \) means "greater than" or "less than".
• \(\geq\) means "greater than or equal to".
• \(\leq\) means "less than or equal to".

These symbols help us understand the relationship between numbers.

In the context of the exercise, \(|x| > 3\) indicates that the solution for \(x\) must be more than 3 units away from 0. This inequality allows us to determine where \(x\) can be on the number line.

Locating Points on the Number Line

A number line is a visual representation of numbers in a straight line. It helps in understanding the position of numbers relative to each other. Let's explore how this applies to the given inequality \(|x| > 3\).

On a number line:

• The number 0 sits at the center.
• Positive numbers are to the right of 0.
• Negative numbers are to the left of 0.

If \(|x| > 3\), it means:

• \(x\) is more than 3 units to the right (\(x > 3\)).
• Or, \(x\) is more than 3 units to the left (\(x < -3\)).

This gives us two regions on the number line where \(x\) can be found, highlighting the flexibility of inequalities in describing a range of values.

Concept of Distance Using Absolute Value

Distance on a number line is often represented using absolute value. This concept reflects how far a number is from 0, without considering direction. For instance, both \(|3|\) and \(|-3|\) equal 3 because they are 3 units from 0.

In the exercise, \(|x| > 3\) signifies that the distance is more than 3 units. Absolute value helps us in:

• Identifying the magnitude of a number.
• Ignoring whether a number is positive or negative.

This way, we focus only on how far away it is, helping us solve problems related to physical and theoretical distances effectively.