Problem 7 - College Algebra with Corequisite Support

Question

Describe all numbers \(x\) that are at a distance of \(\frac{1}{2}\) from the number -4 . Express this set of numbers using absolute value notation.

Step-by-Step Solution

Answer

The set of numbers is \( \left\{ -\frac{7}{2}, -\frac{9}{2} \right\} \).

Step 1: Understanding the Concept of Distance

The problem states that we need to find all numbers that are at a distance of \( \frac{1}{2} \) from \( -4 \). In mathematical terms, distance refers to the absolute value of the difference between two numbers.

Step 2: Setting Up the Absolute Value Equation

We need to express the condition "distance is \( \frac{1}{2} \) from \( -4 \)" using an absolute value equation. This can be represented as:\[ |x + 4| = \frac{1}{2} \] This equation states that the difference between \( x \) and \(-4\) should be \( \frac{1}{2} \).

Step 3: Solving the Absolute Value Equation

To solve \( |x + 4| = \frac{1}{2} \), we break it down into two separate equations because absolute value equations naturally split into positive and negative scenarios:1. \( x + 4 = \frac{1}{2} \)2. \( x + 4 = -\frac{1}{2} \)

Step 4: Solving the First Equation

Solve the equation \( x + 4 = \frac{1}{2} \):Subtract 4 from both sides:\[ x = \frac{1}{2} - 4 \]Simplify:\[ x = \frac{1}{2} - \frac{8}{2} = -\frac{7}{2} \]

Step 5: Solving the Second Equation

Solve the equation \( x + 4 = -\frac{1}{2} \):Subtract 4 from both sides:\[ x = -\frac{1}{2} - 4 \]Simplify:\[ x = -\frac{1}{2} - \frac{8}{2} = -\frac{9}{2} \]

Step 6: Summarizing the Solution

The solutions to the absolute value equation \( |x + 4| = \frac{1}{2} \) are \( x = -\frac{7}{2} \) and \( x = -\frac{9}{2} \). These are the two numbers that satisfy the given condition.

Key Concepts

distance in algebra

In algebra, distance has a unique role, especially when dealing with absolute value equations. Distance measures how far apart two numbers are on the number line. It is always expressed as a non-negative value. This concept can be simplified as the absolute value of the difference between two numbers.
In the provided exercise, the distance of interest is between a variable \( x \) and the fixed number \( -4 \). Given that the distance should be \( \frac{1}{2} \), this is calculated as \(|x + 4|\).
To visualize this, think of distance as the steps needed to reach another point on a number line. Here, it means that \( x \) is either \( \frac{1}{2} \) steps away to the left or to the right of \(-4\), creating two possible solutions to explore.

solving equations

Solving equations is a fundamental skill in algebra where we find the value(s) of the variable(s) that make the equation true. In this exercise, the equation to solve uses absolute value:

• \(|x + 4| = \frac{1}{2}\)

The absolute value creates two scenarios because it measures distance without regard to direction. Thus, the equation \(|a| = b\) splits into two possible equations:

• \(a = b\)
• \(a = -b\)

Applying these principles, our original equation provides two separate cases:

• \(x + 4 = \frac{1}{2}\)
• \(x + 4 = -\frac{1}{2}\)

Each case is then solved algebraically by isolating \(x\). This involves subtracting \(4\) from both sides to find the different values of \(x\) that meet these conditions.

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a particular value. In working with mathematical expressions like equations, translating a word problem into an equation is an essential skill.
In this exercise, expressing the condition 'distance of \(\frac{1}{2}\) from \(-4\)' in mathematical terms required understanding and using an absolute value notation. This transforms a real-world concept into a precise algebraic expression, such as \(|x + 4| = \frac{1}{2}\).
Understanding the components of the equation helps dissect and solve it appropriately. Here, the expression \(x + 4\) suggests adding 4 to \(x\) before evaluating its absolute value, showing how abstract concepts are built through mathematical expressions to reach practical solutions.