Problem 25 - Elementary and Intermediate Algebra

Question

Solve each equation. \(\left|\frac{x}{5}\right|=10\)

Step-by-Step Solution

Answer

The solutions are \( x = 50 \) and \( x = -50 \).

Step 1: Understand the Absolute Value Equation

The given equation is an absolute value equation, \( \left| \frac{x}{5} \right| = 10 \). Recall that \( |A| = B \) means \( A = B \) or \( A = -B \). Thus, here, \( \frac{x}{5} = 10 \) or \( \frac{x}{5} = -10 \).

Step 2: Solve the First Equation

Start with \( \frac{x}{5} = 10 \). Multiply both sides by 5 to solve for \( x \). This gives \( x = 10 \times 5 = 50 \).

Step 3: Solve the Second Equation

Now solve \( \frac{x}{5} = -10 \). Multiply both sides by 5 to solve for \( x \). This gives \( x = -10 \times 5 = -50 \).

Step 4: Solution Set

The solutions to the equation \( \left| \frac{x}{5} \right| = 10 \) are \( x = 50 \) and \( x = -50 \).

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics. It typically involves finding the value or values of the variable that make the equation true. In this exercise, we are dealing with an absolute value equation. When solving such equations, it's important to understand that the absolute value symbol causes the expression inside to become positive.
For the given equation \( \left| \frac{x}{5} \right| = 10 \), this means we can have two scenarios to consider:

• The expression inside the absolute value equals 10, thus \( \frac{x}{5} = 10 \).
• The expression can also equal -10, thus \( \frac{x}{5} = -10 \).

The goal is to solve each of these equations separately. For both equations, we solve for \( x \) by multiplying each side by 5. This technique clears the fraction and yields the solution for \( x \). In this way, solving an absolute value equation involves considering multiple cases to find all possible solutions.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between numbers and variables using algebraic operations such as addition, subtraction, multiplication, and division.
In our exercise, \( \frac{x}{5} = 10 \) and \( \frac{x}{5} = -10 \) are algebraic equations derived from the absolute value context. The challenge is to isolate \( x \) in these linear equations, which is accomplished through careful algebraic manipulation.

• Start with one of the equations such as \( \frac{x}{5} = 10 \).
• To eliminate the fraction, multiply both sides by 5, giving \( x = 50 \).
• Repeat this process for \( \frac{x}{5} = -10 \) to find the second solution: \( x = -50 \).

These steps demonstrate how fundamental arithmetic operations play a crucial role in solving algebraic equations. By consistently applying these methods, you ensure clarity and accuracy in finding the solutions.

High School Mathematics

High school mathematics covers a wide range of topics, from basic arithmetic to more advanced subjects like algebra and calculus. Understanding how to work with equations, especially absolute value equations, is a key skill at this level.
The exercise here illustrates typical problem-solving techniques you might encounter in high school algebra. Not only does it include understanding what an absolute value represents, but it also involves using algebraic strategies to solve linear equations.

• The concepts of absolute values teach you how to handle quantities that disregard sign, emphasizing the value alone.
• Working through cases like \( \frac{x}{5} = 10 \) and \( \frac{x}{5} = -10 \) helps reinforce solving linear equations through balancing and isolation techniques.

Mastering these skills prepares students for more complex mathematical explorations, setting a foundation for solving diverse problems effectively in academic and real-world scenarios.