Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Use absolute value notation to describe the sentence.\(x\) is more than five units from \(m\).

Step-by-Step Solution

Answer

The mathematical expression to represent the statement '\(x\) is more than five units from \(m\)' using absolute value notation is \(|x-m|>5\).

Step 1: Understand the Question

We are asked:
Use absolute value notation to describe the sentence.\(x\) is more than five
units from \(m\).

Step 2: Recall the Definition

We recall the relevant mathematical definition or concept.

Step 3: State the Answer

The mathematical expression to represent the statement '\(x\) is more than five units from \(m\)' using absolute value notation is \(|x-m|>5\).

Key Concepts

Distance

Distance in mathematics can often be thought of as the amount of space between two points. When we use absolute value, it gives us the non-negative measure of this space, which is the distance between two numbers on a number line.

• For example, the distance between the numbers 3 and 8 is 5 units. We calculate it as \(|8-3| = 5|\).
• The absolute value ensures the result is always a positive number because distance cannot be negative.
• It’s a valuable tool in assessing how far apart two values are, without concern for direction or negative results.

This concept becomes handy when we translate sentences like 'x is more than five units from m.' We simply measure the distance between x and m and check if it is more than 5, using the absolute value notation, \(|x-m|>5\). This effectively describes all possible values of x that are more than 5 units away from m.

Inequalities

Inequalities allow us to make comparisons between two expressions or values. When dealing with absolute values, we use inequalities to express the range of possible values that fulfill a specific condition.
A sentence like 'x is more than five units from m' translates to the absolute value inequality \(|x-m|>5\).

• This inequality means that the distance between x and m should be greater than 5.
• In simpler terms, x should be at least 6 units to the left or to the right of m.

To solve this inequality, we consider two separate cases because absolute value inequalities like \(|x-m|>5\) turn into two potential inequalities:

• Either \(x-m>5\) (x is more than 5 units away on the positive side), or
• \(x-m<-5\) (x is more than 5 units away on the negative side).

Thus, inequalities in this context help define the conditions under which the absolute value expression holds.

Mathematical Translation

Mathematical translation involves converting verbal statements into mathematical expressions or equations. This process is crucial to solve word problems successfully. When given a sentence such as 'x is more than five units from m,' it's important to identify key words and phrases that indicate mathematical operations or concepts.
In this case:

• 'More than five units' suggests a distance, which we know from absolute value concepts.
• 'Is more than' points to an inequality.

So, the statement translates into an absolute value inequality, \(|x-m|>5\).Here is a quick guide on how to effectively translate sentences into mathematical expressions:

• Identify keywords that refer to mathematical operations (e.g., 'more than' for inequalities).
• Recognize terms indicating the distance or absolute value.
• Write down the corresponding mathematical notation that captures the idea conveyed by the statement.

Mastering mathematical translation bridges the gap between language and math, making it easier to tackle various problems.