Problem 97 - 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 2 .

Step-by-Step Solution

Answer

The sentence 'The distance from \(0\) to \(x\) on a number line is greater than \(2\)' translates to the inequality \(|x| > 2\) in mathematical terms. The possible locations of \(x\) are all the values less than \(-2\) or greater than \(2\).

Step 1: Interpret the English sentence mathematically

When we say 'the distance from 0 to \(x\) on a number line is greater than \(2\)', it means \(x\) is more than \(2\) units away from \(0\) in both positive and negative direction.

Step 2: Rewrite the sentence as an absolute value inequality

Since distance cannot be negative and distance greater than 2 from origin (0) is in both direction, this can be expressed with the absolute value as \(|x| > 2\).

Step 3: Find the possible locations of \(x\)

The inequality \(|x| > 2\) on the number line means \(x\) can be less than \(-2\) or greater than \(2\). In other words, \(x\) can be anywhere on the number line except between -2 and 2.

Key Concepts

Distance on a Number Line

Understanding the concept of distance on a number line is crucial, especially when dealing with absolute values. In mathematics,

• distance refers to how far a point is from another point.
• On a number line, the distance from a number to zero is easily visualized by the positioning of these numbers.

When you're asked to find 'the distance from 0 to \(x\) on a number line', using absolute value notation becomes highly practical. Absolute value, represented by \(|x|\), measures this distance without considering direction. So, whether a number is to the right (positive) or to the left (negative) of zero, the distance is always non-negative and straightforward to calculate. For example:

• When \( x = 3 \), its distance from 0 is \(|3| = 3\).
• Similarly, when \( x = -5 \), its distance from 0 remains the same, \(|-5| = 5\).

These properties make absolute value inequalities especially useful in describing the range and restriction on the number line that \(x\) might occupy.

Inequalities

Inequalities are expressions that define the relationship between two values when they're not equal. They are important in identifying the set of possible solutions within a specific range. Using inequalities, we can specify regions on the number line where a particular condition is fulfilled. When we transform statements like, 'the distance from 0 to \(x\) is greater than 2', into a mathematical inequality, it becomes \(|x| > 2\). This expression tells us that the absolute value, or the absolute distance of \(x\) from zero, needs to exceed 2 units on the number line.

• These inequalities can represent a wide range of solutions.
• In this case, \(x\) must be more than 2 units to the right or left of zero.

It's also key to recognize how inequalities are solved, often involving solving them in two segments: one representing positive distances (such as \(x > 2\)) and the other representing negative distances (like \(x < -2\)). This dual approach accurately captures all solutions for \(x\) on the number line.

Number Line Concepts

Number lines are visual tools used to depict numerical relationships and order. They help show the exact position of numbers in a sequence. One of their most frequent uses is for visualizing inequalities and absolute values. When numbers are plotted on a number line, it becomes easy to see:

• how far a number is from another number (like zero)
• what segment of the number line a solution for an inequality might occupy

For the inequality \(|x| > 2\), the number line perspective can visually illustrate the values \(x\) can take:

• All values less than -2 (\(x < -2\))
• And all values more than 2 (\(x > 2\))

This visual segmentation shows that solutions lie in two distinct sections of the line, clear and separate from each other. Hence, number line visuals make abstract mathematical concepts tangible and more accessible. Understanding how to read and plot on a number line is fundamental for grasping more complex mathematical ideas flexibly and thoroughly.