Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\mathrm{To}\) _______ the absolute value in \(|3-x|-4=5,\) we add 4 to both sides.

Step-by-Step Solution

Answer

Add 4 to both sides.

Step 1: Identify the Equation

We start with the given equation \(|3-x|-4=5\). Our objective is to isolate the absolute value expression \(|3-x|\).

Step 2: Isolate the Absolute Value

To isolate \(|3-x|\), we reverse the subtraction by adding 4 to both sides of the equation: \(|3-x|-4 + 4 = 5 + 4\). This simplifies to \(|3-x| = 9\).

Key Concepts

Absolute Value

The concept of absolute value is a fundamental building block in algebra. Simply put, the absolute value of a number is its distance from zero on a number line, without considering direction. Absolute values are always non-negative. For example, the absolute value of both 6 and -6 is 6 because both are 6 units away from zero. This is represented as \(|6| = 6\) and \(|-6| = 6\).

When dealing with algebraic expressions like \(|3-x|\), the absolute value indicates that you're interested in the size of the expression, rather than its sign. In equations, it's crucial to handle absolute values carefully to ensure solutions are valid. This can involve splitting equations into two separate cases, one where the expression inside is non-negative, and another where it is negative. However, before making such cases, it is essential to first isolate the absolute value in the equation.

Isolating Expressions

Isolating expressions is a key step in solving algebraic equations, particularly those involving absolute values or other complex structures. This process involves manipulating the equation to 'isolate' the particular term or expression of interest on one side of the equation.

Consider the equation \(|3-x|-4=5\). The initial form can be challenging to work with directly because the absolute value expression \(|3-x|\) is not isolated. To isolate this expression, we need to perform operations that will set it apart on one side. In this example, adding 4 cancels the subtraction, simplifying the equation to \(|3-x| = 9\).

Always remember, whatever mathematical operation you do to one side of an equation, you must do to the other side as well. This maintains the balance of the equation, which is crucial when isolating expressions. The goal is to make the variable or expression the focus, to better analyze and solve the equation.

Equations

Equations serve as the core language of algebra, representing equalities that contain variables. Solving equations involves finding the value of the variables that make the equation true. They come in many forms and complexities.

The example equation \(|3-x|-4=5\) began with an absolute value expression. By isolating this expression, we transform the equation into a simpler form \(|3-x| = 9\).

At this point, you tackle the problem by considering both possible scenarios for the value inside the absolute value: \(3-x=9\) and \(3-x=-9\). Solving these will give you the possible solutions to the equation.

• In \(3-x=9\), solving for \(x\) gives \(x=-6\).
• In \(3-x=-9\), solving for \(x\) gives \(x=12\).

By testing these solutions back in the original equation, you can verify their accuracy. Therefore, understanding the process of solving equations, especially those involving absolute values, helps reveal all possible solutions and ensures none are overlooked.