Problem 31 - Intermediate Algebra

Question

Solve each equation. See Example 2. $$ |x-3|-19=3 $$

Step-by-Step Solution

Answer

The solutions are \( x = 25 \) and \( x = -19 \).

Step 1: Isolate the Absolute Value

First, add 19 to both sides of the equation to isolate the absolute value expression: \( |x - 3| - 19 = 3 \) becomes \( |x - 3| = 22 \).

Step 2: Set Up Two Equations

An absolute value equation \( |A| = B \) implies two possible equations: \( A = B \) and \( A = -B \). So, set up the equations\( x - 3 = 22 \) and \( x - 3 = -22 \).

Step 3: Solve the First Equation

Solve \( x - 3 = 22 \) by adding 3 to both sides to get \( x = 25 \).

Step 4: Solve the Second Equation

Solve \( x - 3 = -22 \) by adding 3 to both sides to get \( x = -19 \).

Step 5: Write the Solution Set

The solutions from the two equations are \( x = 25 \) and \( x = -19 \). Therefore, the solution set is \( \{25, -19\} \).

Key Concepts

Solving Equations

When solving equations, we are looking for the value of the variable that makes the equation true. In the realm of algebra, equations can take many forms, but one of the types that often poses a challenge is the absolute value equation. Absolute value equations contain expressions within absolute value bars. The absolute value measures how far a number is from zero on the number line. It’s important to understand that the absolute value of a number is always non-negative.
\[ |A| = B \]
To solve an absolute value equation like \(|x - 3| = 22\), follow these steps:

• Isolate the absolute value expression if it's not already. In our case, this was achieved by adding 19 to both sides, giving us \(|x - 3| = 22\).
• Once isolated, set up two separate equations based on the absolute value property: one where the expression inside the absolute value is equal to the positive number, and one where it equals the negative of that number. Thus, \(x - 3 = 22\) and \(x - 3 = -22\).
• Solve each resulting equation separately. Solving both equations gives us the values of the variable that will satisfy the original equation.

By following these steps, you can systematically find all possible solutions for the absolute value equation.

Solution Set

The solution set of an equation includes all values of the variable that satisfy the equation. For absolute value equations, due to their nature, there can be two solutions. This is because the absolute value can represent a positive or negative counterpart of the same number. Thus, when you isolate the absolute value and set up two separate equations, you effectively expand your search for solutions.

• After solving the first equation, \(x = 25\), and solving the second equation, \(x = -19\), you compile these solutions into what we call the solution set.
• It's often represented as a set of numbers, usually enclosed within curly braces. For our equation \(|x-3|-19=3\), the solution set is \(\{25, -19\}\).

The key here is understanding that both values are correct as they both satisfy the original equation once new conditions are set and solved. Remember, in cases of absolute value equations, there could be two values in the solution set, just like in this example.

Algebraic Expressions

Algebraic expressions are the building blocks of algebraic equations. They include variables, numbers, and operational symbols. In the equation \(|x - 3| = 22\), the expression inside the absolute value is \(x - 3\).

Algebraic expressions can be manipulated in various ways to isolate and solve for variables. This manipulation involves operations such as addition, subtraction, multiplication, and division. These operations allow us to transform equations and simplify expressions in order to find a solution.

• When solving absolute value equations, like \(|x - 3| = 22\), focus first on manipulating the expression to isolate variables. Recognizing that the expression \(x-3\) can have two outcomes is crucial.
• In our example, by considering both \(x - 3 = 22\) and \(x - 3 = -22\), we treat these expressions as two separate, basic algebraic equations to be solved.

Understanding how to manipulate and solve algebraic expressions is essential for any algebra problem, particularly when dealing with more complex equations such as those involving absolute values.