Problem 6 - College Algebra

Question

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

Step-by-Step Solution

Answer

The numbers are \( x = -\frac{7}{2} \) and \( x = -\frac{9}{2} \).

Step 1: Understanding the Problem

We need to find all numbers \( x \) that are at a distance of \( \frac{1}{2} \) from the number \( -4 \). This means that the difference between \( x \) and \( -4 \) is \( \frac{1}{2} \).

Step 2: Setting Up the Equation

The distance \( |x - (-4)| \) from \( x \) to \(-4\) is given to be \(\frac{1}{2}\), so we set up the equation: \( |x + 4| = \frac{1}{2} \).

Step 3: Solving the Absolute Value Equation - First Case

The equation \( |x + 4| = \frac{1}{2} \) can be solved by considering two cases. In the first case, \( x + 4 = \frac{1}{2} \). Solve for \( x \) to get \( x = \frac{1}{2} - 4 = -\frac{7}{2} \).

Step 4: Solving the Absolute Value Equation - Second Case

In the second case, \( x + 4 = -\frac{1}{2} \). Solve for \( x \) to get \( x = -\frac{1}{2} - 4 = -\frac{9}{2} \).

Step 5: Conclusion - Listing Solutions

The numbers that satisfy the condition are \( x = -\frac{7}{2} \) and \( x = -\frac{9}{2} \). These are the numbers that are at a distance of \( \frac{1}{2} \) from \(-4\).

Key Concepts

distance in algebra

Sometimes, in algebra, we use the concept of "distance" to describe how far one number is from another on the number line. This is where absolute values come into play.
The absolute value of a number is the magnitude of that number regardless of its sign. It's like counting how many steps you take, and it doesn't matter if your steps take you into the negative or positive part of the number line.

• When you're calculating the distance between two numbers, such as "being a distance of \( \frac{1}{2} \) from -4", you are thinking of both numbers \( x \) which are that distance away - whether slightly bigger or smaller than -4.
• This can be represented as \( |x - (-4)| = \frac{1}{2} \).

This tells us that when we calculate how many steps \( x \) is from \(-4\), the result is \( \frac{1}{2} \). Using this understanding is key to work through problems like these as it sets the stage for solving equations.

solving equations

Once we have interpreted the absolute value expression, the next step is solving for \( x \). Absolute value equations such as \(|x + 4| = \frac{1}{2}\) involve looking at more than one possibility, because when the absolute value of a number is given, that number can be either positive or negative.
Here's how we solve it:

• Begin by removing the absolute value and setting up two separate equations since the expression inside the absolute value could be equal to the given positive distance or its negative:
• For the first equation: \( x + 4 = \frac{1}{2} \)
• For the second equation: \( x + 4 = -\frac{1}{2} \)

This step-by-step breakdown allows us to solve each possible case separately and find all possible solutions. Once each scenario is solved, you check each solution to ensure they fit within the originally described problem context.

algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like addition or subtraction). In expressions like \( |x + 4| = \frac{1}{2} \), \( x \) represents our unknown and is what we are solving for.

• The expression \( x + 4 \) is called a "linear expression", where \( x \) is the variable whose value changes and affects the whole expression.
• By manipulating the algebraic expression, such as adding or subtracting numbers to isolate \( x \), we essentially solve the equation.
• These manipulations demonstrate the beauty of algebra, transforming a seemingly complex statement into two simple arithmetic problems that we can solve in quick steps.

Each part of an algebraic expression has its function, helping us define relationships between numbers and solve for unknown values efficiently.