Problem 127 - Elementary and Intermediate Algebra

Question

What numbers are a distance of 6 away from \(-11\) on a number line?

Step-by-Step Solution

Answer

The numbers are -5 and -17.

Step 1: Understand the Distance Concept

On a number line, the distance between two points is the absolute difference of their values. We are looking for numbers that are exactly 6 units away from -11.

Step 2: Set Up the Equation

If a number is 6 units away from -11, then the absolute value of the difference between that number (let's call it 'x') and -11 is 6. This can be set up in the following equation: \[ |x + 11| = 6 \]

Step 3: Solve the Absolute Value Equation

The equation \( |x + 11| = 6 \) results in two separate linear equations because absolute value equations are solved by considering both the positive and negative cases. The two equations are:1. \( x + 11 = 6 \)2. \( x + 11 = -6 \)

Step 4: Solve the First Equation

Solve the equation \( x + 11 = 6 \) by isolating x:\[\begin{align*}x + 11 &= 6 \\Rightarrow x &= 6 - 11 \\Rightarrow x &= -5\end{align*}\]

Step 5: Solve the Second Equation

Solve the equation \( x + 11 = -6 \) by isolating x:\[\begin{align*}x + 11 &= -6 \\Rightarrow x &= -6 - 11 \\Rightarrow x &= -17\end{align*}\]

Step 6: Determine the Numbers on the Number Line

The solutions to the equations give us the numbers that are 6 units away from -11 on the number line. These numbers are -5 and -17.

Key Concepts

Number Line

A number line is a straight line that visually represents numbers in order. It functions as a powerful tool to understand number positions and relationships. The line is marked with integers at equal intervals, extending infinitely in both directions. Zero is typically found at the center, with positive numbers to the right and negative numbers to the left.

The usefulness of a number line lies in its capability to illustrate the concept of distance between numbers. For instance, if you are looking to understand what numbers are a set distance away from a particular point, like -11, a number line helps clarify which points fall within that specific range of units.

This visual tool aids in understanding basic operations, including addition and subtraction, by moving right or left from a specific point. By showing us how far apart numbers are, number lines can help demonstrate concepts like absolute value, making abstract mathematical concepts more concrete.

Solving Linear Equations

Linear equations are mathematical statements that describe a straight-line graph. They typically take the form of \( ax + b = c \), where \( x \) is the unknown variable we need to solve for. Solving linear equations involves isolating this variable to find its value.

In the context of our exercise, we face an absolute value equation: \( |x + 11| = 6 \). Absolute value equations break into two linear equations because absolute value measures only distance, regardless of direction. One equation simulates the positive distance case, and the other simulates the negative.

This division into two simple linear equations:\

\
• \( x + 11 = 6 \)
\
• \( x + 11 = -6 \)
\

allows us to find the exact values that solve the absolute value condition. Each equation can be solved by step-by-step transpositions, leading to isolating \( x \) and hence finding our solutions: \( x = -5 \) and \( x = -17 \).

Distance on a Number Line

Distance on a number line refers to the difference in position between two points. It is represented by the absolute value of their difference, providing a non-negative measure of separation. This concept allows us to determine how far one number is from another.

When given a problem involving a set distance from a point, such as finding numbers exactly 6 units away from -11, you utilize the principle of absolute difference. Such a challenge leads naturally to the formation of an absolute value equation like \( |x + 11| = 6 \).

The response to that setup is all about enumerating two possibilities analyzing both directions along the number line. Solving gives us two solutions: numbers that lie equally distanced from -11 are -5 and -17. These solutions remind us how absolute value captures our sense of distance in mathematics—directionless but straightforward, mirroring reality's consistent numerical gaps.