Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Use absolute value notation to describe the sentence.\(y\) is at most two units from \(a\).

Step-by-Step Solution

Answer

The sentence 'y is at most two units from a' in the absolute value notation is \(|y - a| \leq 2\)

Step 1: Understand the Concept of Absolute Value

Understanding that the absolute value of a number represents its distance from zero on a number line. Therefore in our case, \(|y - a|\), will indicate the distance of y from a.

Step 2: Translate the Phrase into Mathematical Statement

Next, translate the sentence 'y is at most two units from a'. The phrase 'at most' will translate mathematically into 'less than or equal to'. Hence the translated statement will be \(|y - a| \leq 2\).

Step 3: Finalize the Mathematical Notation

Combine all to finalize the mathematical notation of the given sentence, the final result will be \(|y - a| \leq 2\)

Key Concepts

Distance on a Number Line

When we talk about the 'distance on a number line', it's all about understanding how far numbers are from each other. Imagine a straight line marked with numbers, like a ruler. Each point on this line represents a number.

To find the distance between two points, you focus on how many units or steps apart they are. If you have two points, let's call them \(y\) and \(a\). The absolute value notation \(|y - a|\) is used to express their distance.

Remember, distance is always positive, because it measures how far apart two points are. So if \(y\) is 3 units in front of \(a\) or behind it, the distance is still 3 units.

• Distance doesn't care about direction, just the gap between numbers.
• It's a fundamental idea for many mathematical concepts.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values in terms of size. It's like saying one number can be bigger, smaller, or sometimes equal to another.

For example, when talking about "at most two units from \(a\)," it translates to using \(\leq 2\) in mathematical language. This means the distance can't be more than 2 units.

In this context, \(|y - a| \leq 2\) means the absolute distance between \(y\) and \(a\) should be 2 or less. This is how you express limits and boundaries mathematically.

• "\(<\)" means less than, while "\(>\)" means greater than.
• "\(\leq\)" and "\(\geq\)" include equality.

Mathematical Expressions

Mathematical expressions are like the way we speak numbers. They combine numbers, variables, and operations to form meaningful sentences in math.

Expressions like \(|y - a| \leq 2\) tell a story: they describe the relationship or condition between \(y\) and \(a\). Using variables, you make statements that can apply to many numbers or situations.

The "+", "-", "*", and "/" are basic operators that we often see. When combined with numbers and variables, they help form expressions that can describe real-world situations.

• Expressions don't always need to have an equals sign \((=)\).
• Think of them as formulas or equations waiting for more information.