Problem 96 - Introductory and Intermediate Algebra for College Students 4th

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 less than 3 .

Step-by-Step Solution

Answer

The sentence 'The distance from 0 to \(x\) on a number line is less than 3' translates into the inequality \(|x| < 3\) which also can be broken down into \(-3 < x < 3\).

Step 1: Understanding Absolute Value

Absolute value of a number \(|x|\) gives us the distance from the number to 0 on a number line. It doesn't matter if the number is to the left or right of zero (i.e., negative or positive), the absolute value always returns a positive value or zero.

Step 2: Translating the Sentence into Mathematical terms

The sentence 'The distance from 0 to \(x\) on a number line is less than 3' can be translated into an absolute value inequality as follows: The distance from 0 to \(x\), or \(|x|\), is less than 3, can be written as the inequality \(|x| < 3\). This means \(x\) can be any number that is less than 3 units away from zero on the number line.

Step 3: Breaking down the Inequality

The absolute value inequality \(|x| < 3\) represents two separate inequalities if we break it down. The number \(x\) should be less than 3 and greater than -3, as \(x\) is less than 3 units away from zero on either sides, which gives us: \(-3 < x < 3\). This represents all the possible locations for \(x\) on the number line as described by the initial sentence.

Key Concepts

number line

A number line is a straightforward way to visualize numbers in a linear format. It has zero at the center, negative numbers extending to the left, and positive numbers extending to the right.

By using a number line, we can easily locate and compare numbers, as well as perform operations like addition and subtraction. For example, -2 would be two units to the left of zero, while 4 would be four units to the right of zero.

• Visualizes the position of numbers
• Makes comparisons effortless
• Facilitates arithmetic operations

Understanding a number line is crucial, especially when dealing with absolute values and inequalities. It offers a clear representation of where numbers lie in relation to zero, which is an essential concept in mathematical expressions.

distance from zero

The distance from zero is a key concept when working with absolute values. It tells us how far a number is from zero on a number line, ignoring whether the number is positive or negative.

This is why the absolute value is always non-negative. For instance, both -3 and 3 are three units away from zero, meaning \(|-3| = 3\) and \(|3| = 3\). Understanding this distance helps in solving absolute value inequalities by clearly indicating the range within which a number can be from zero.

• Is independent of direction
• Always non-negative
• Crucial for solving inequalities

Absolute values offer a simple way to describe conditions like being 'less than 3 units away from zero', as seen in the inequality representation.

inequality representation

Inequality representation allows us to express ranges or limits on the number line. It shows how absolute value conditions translate into these limits.

For example, the expression \(|x| < 3\) means that x can be any value that is less than 3 units away from zero. This inequality can be split into two: \-3 < x < 3\.

• Denotes a range on the number line
• Involves less than, greater than, or equal conditions
• Essential for defining boundaries

By translating an absolute value inequality into a simple inequality, we define a precise segment on the number line, ensuring students understand the allowed range for x.

translating verbal phrases into mathematical expressions

Translating verbal phrases into mathematical expressions is a skill that enables you to turn everyday language into a form ready for mathematical analysis. The phrase "the distance from 0 to \(x\) on a number line is less than 3" can be expressed as the inequality \(|x| < 3\).

This process involves understanding the meaning behind words and converting them into mathematical symbols and notation.

• Transforms language into math symbols
• Clarifies the problem statement
• Aids in solving complex problems

It's vital for solving real-world problems where conditions need to be interpreted and calculated mathematically. Mastering this skill ensures that the initial verbal condition is accurately represented as an efficient equation or inequality on a number line.