Problem 19 - Algebra 2 and Trigonometry

Question

Two distinct points on the number line represent the numbers \(a\) and \(b\) . If \(|5-a|=|5-b|=6,\) what are the values of \(a\) and \(b ?\)

Step-by-Step Solution

Answer

The values of \(a\) and \(b\) are -1 and 11.

Step 1: Identifying Absolute Value Equation Conditions

We start with the given equation \(|5-a| = 6\). This equation means the distance between 5 and \(a\) on the number line is 6. The absolute value equation \(|x - c| = d\) has two solutions: \(x - c = d\) or \(x - c = -d\).

Step 2: Solving for 'a'

For \(|5-a| = 6\), we set up two cases: 1. \(5 - a = 6\) which simplifies to \(a = -1\).2. \(5 - a = -6\) which simplifies to \(a = 11\).So, the possible values for \(a\) are -1 and 11.

Step 3: Solving for 'b'

Similarly, for \(|5-b| = 6\), we set up two cases:1. \(5 - b = 6\) which simplifies to \(b = -1\).2. \(5 - b = -6\) which simplifies to \(b = 11\).So, the possible values for \(b\) are -1 and 11.

Step 4: Identifying Distinct Values for 'a' and 'b'

We know \(a\) and \(b\) must be distinct. From previous calculations, possible values for both \(a\) and \(b\) are -1 and 11. Thus, \(a = -1\) and \(b = 11\) or \(a = 11\) and \(b = -1\). Either pair gives distinct values for \(a\) and \(b\).

Key Concepts

Number Line

Think of a number line as a straight path. It contains all real numbers, lined up left to right or right to left. Each spot on this line corresponds to a number, which you can think of like addresses.

Here are a few key points to remember when working with a number line:

• Zero typically acts as the center, with positive numbers on the right and negative numbers on the left.
• Numbers get bigger as you move to the right and smaller as you move to the left.
• It's often used to visualize the concepts of addition and subtraction as movement along this line.

For the equation \(|5-a|=6\), you find a spot called 5 on this line. Then, you measure 6 spaces away to discover possible positions for point \(a\). This illustration can make complex math, like absolute values, clearer.

Distinct Points

In math, distinct points refer to spots that are separate and different from each other on the number line.

Let’s see how that works:

• If you have points \(a\) and \(b\), these might fall in different places unless stated otherwise.
• In our problem, \(a\) and \(b\) must not only satisfy the conditions of the equation but must also be distinct. This means \(a\) and \(b\) cannot be the same number.

When solving math problems, keeping track of distinct numbers is crucial. For example, when you end up with possible values like \(-1\) and \(11\), ensuring that one point takes on \(-1\) and the other \(11\) fulfills the condition we are looking for in distinct points. It’s a reminder that finding possible solutions isn’t just about solving equations; it’s also about respecting the constraints of being unique or distinct.

Distance on the Number Line

Distance on a number line is simply the space between two numbers, regardless of direction. This distance is critical in understanding absolute values.

Here’s what you should know:

• The distance between two points \(a\) and \(b\) is calculated as \(|a-b|\).
• The absolute value \(|x-c|=d\) refers to a point \(x\) being \(d\) units away from \(c\) on the number line.
• Absolute value equations give you positive distance, never negative, as they measure how far apart numbers are.

This is why when you see \(|5-a|=6\), it translates to point \(a\) being 6 units away from 5, offering two possible distance points on either side of 5, like \(-1\) and \(11\). This concept helps immensely in visualizing and solving similar equations.

Solving Equations

Solving equations is like peeling an onion. You peel back layers to reveal the core solution. In the case of absolute value equations, you’ll solve by considering two scenarios due to the positive distance interpretation.

Here's how it breaks down:

• An equation \(|5-a|=6\) translates to two straightforward cases: \(5-a=6\) or \(5-a=-6\).
• Each equation is then solved separately: solving \(5-a=6\) yields \(a=-1\), while \(5-a=-6\) yields \(a=11\).
• This method applies the same way when you are solving for another variable like \(b\).

It's critical to check your solutions to ensure they make sense within the context and constraints provided, like ensuring values are distinct when needed. Solving equations often gives you foundational skills not only for resolving immediate problems but better understanding math structures as a whole.