Problem 6 - 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 numbers are \(-\frac{7}{2}\) and \(-\frac{9}{2}\).

Step 1: Understanding the Problem

We need to find all numbers that are at a distance of \( \frac{1}{2} \) from \(-4\). Mathematically, this means we need to find numbers \(x\) such that the distance between \(x\) and \(-4\) is \(\frac{1}{2}\).

Step 2: Set up the Absolute Value Equation

The distance from \(x\) to \(-4\) is expressed as \(|x - (-4)| = |x + 4|\). We want this distance to be \(\frac{1}{2}\). Thus, the equation is \(|x + 4| = \frac{1}{2}\).

Step 3: Solve the Absolute Value Equation

We know that \(|x + 4| = \frac{1}{2}\) means there are two possible solutions: \(x + 4 = \frac{1}{2}\) or \(x + 4 = -\frac{1}{2}\).

Step 4: Solve Each Equation

1. Solve \(x + 4 = \frac{1}{2}\): - Subtract 4 from each side: \(x = \frac{1}{2} - 4 = -\frac{7}{2}\).2. Solve \(x + 4 = -\frac{1}{2}\): - Subtract 4 from each side: \(x = -\frac{1}{2} - 4 = -\frac{9}{2}\).

Step 5: State the Solution in a Set

The set of numbers \(x\) that are at a distance of \(\frac{1}{2}\) from \(-4\) is \( \left\{ -\frac{7}{2}, -\frac{9}{2} \right\} \).

Key Concepts

Distance from a Number

Understanding the concept of distance mathematically is crucial, especially when dealing with absolute value equations. The distance between two numbers on a number line is essentially the absolute difference between them. In simpler terms, if you have two numbers like \( x \) and \( -4 \), the distance is how far \( x \) is from \( -4 \) on the number line.
This distance is always a non-negative value, which is where absolute value comes into play. Absolute value makes sure we ignore the direction on the number line and focus purely on magnitude.
So when the exercise asks about numbers "at a distance of \( \frac{1}{2} \) from \(-4\)", it translates into an absolute value equation. The statement "at a distance of\( \frac{1}{2} \)" gets mathematically expressed as \(|x + 4| = \frac{1}{2}\). The "\(+4\)" in \(|x + 4|\) comes from recognizing that measuring from \(-4\) means adding 4 within the equation.
This step transforms the word problem into a solvable math equation.

Solution Sets

The solution set in an equation refers to all possible values that satisfy the equation. In other words, these are the values of \( x \) that make the equation true. When you solve absolute value equations, you're typically looking for all numbers that can lie within a certain distance from a specified point.
In this case, our equation was \(|x + 4| = \frac{1}{2}\).

• The absolute value means that \( x + 4 \) can either be positive \( \frac{1}{2} \) or its negative \(-\frac{1}{2}\).
• Thus, the equation can split into two separate equations: \( x + 4 = \frac{1}{2} \) and \( x + 4 = -\frac{1}{2} \).

Solving these two resulting linear equations led us to find two potential values for \( x \), forming our solution set.
The result is exactly \( \left\{ -\frac{7}{2}, -\frac{9}{2} \right\} \) when solved step-by-step, which will perfectly tell you all calculated distances that lie precisely \( \frac{1}{2} \) away from \(-4\) on the number line.

Solving Equations Step by Step

Breaking down equations step-by-step is a key strategy in mathematics. It not only simplifies your approach but also ensures accuracy in finding solutions. When dealing with absolute value equations, follow these steps:

• First, understand the absolute value equation: Know what the absolute equation \(|x + 4| = \frac{1}{2}\) represents. Here it denotes two scenarios: \( x + 4 = \frac{1}{2} \) and \( x + 4 = -\frac{1}{2} \).


• Next, isolate \( x \) in each equation: Solve both linear equations separately by moving constants to one side.
  • For \( x + 4 = \frac{1}{2} \), subtract 4 from both sides: \( x = -\frac{7}{2} \).
  • For \( x + 4 = -\frac{1}{2} \), again subtract 4: \( x = -\frac{9}{2} \).


• Verify your solutions: Make sure each result actually makes \(|x + 4| = \frac{1}{2}\) true.

By following these steps, you will achieve a complete and correct solution set, enabling a sophisticated understanding of the mathematical problem at hand.