Problem 25 - Intermediate Algebra

Question

Describe the set graphed below using interval notation. $$ \left|\frac{x}{5}\right|=10 $$

Step-by-Step Solution

Answer

The set is \( \{-50, 50\} \) in interval notation.

Step 1: Solve the Equation

The equation given is \( \left| \frac{x}{5} \right| = 10 \). Start by setting up two equations from the absolute value expression: \( \frac{x}{5} = 10 \) and \( \frac{x}{5} = -10 \). This is because an absolute value equal to 10 means that either \( \frac{x}{5} \) is 10 or -10.

Step 2: Solve for x

First, solve the equation \( \frac{x}{5} = 10 \). Multiply both sides by 5 to get \( x = 50 \). Next, solve the equation \( \frac{x}{5} = -10 \). Again, multiply both sides by 5 to get \( x = -50 \). This gives us the solutions \( x = 50 \) and \( x = -50 \).

Step 3: Write the Solution in Interval Notation

The solutions from Step 2 are \( x = 50 \) and \( x = -50 \). In interval notation, these are represented individually as \( \{-50, 50\} \) because they are specific points on the number line, rather than a range of values.

Key Concepts

Absolute Value Equations

Understanding absolute value equations is crucial in algebra as these are foundational in mathematical problem-solving. The absolute value of a number represents its distance from zero on the number line, disregarding any negative signs. When you see an absolute value equation like \( \left| \frac{x}{5} \right| = 10 \), you need to determine what values of \( x \) make this true.

In essence, an equation involving absolute value signifies that the expression within the absolute value (\( \frac{x}{5} \) in this case) can either be 10 or -10. Thus, you have two different scenarios to solve. This results in two linear equations: \( \frac{x}{5} = 10 \) and \( \frac{x}{5} = -10 \). By understanding this concept, you can tackle a wide array of equations and deepen your understanding of both positive and negative number interactions.

Solving Equations

Once you've grasped the nature of absolute value equations, the next step is to solve the resulting linear equations. Let's break it down for the given equation:

• Start with the absolute value expression being set equal to a positive and a negative number: \( \frac{x}{5} = 10 \) and \( \frac{x}{5} = -10 \).
• To isolate \( x \), you will want to eliminate the fraction. Multiply both sides of each equation by 5 to get rid of the denominator.
• This simplifies the equations to \( x = 50 \) and \( x = -50 \).

These steps exemplify systematic thinking and are a regular process in solving equations. By approaching each problem methodically—isolating the variable and simplifying—you can successfully resolve a wide range of mathematical challenges.

Interval Notation Examples

After solving the equations, it is common practice to express the solution set using interval notation. This form of notation efficiently describes ranges or specific values on the number line. Let’s examine how it’s applied here:

• Since our solutions are specific and not intervals (i.e., individual points on the number line, namely \( x = 50 \) and \( x = -50 \)), interval notation uses curly braces to list these points.
• Therefore, the interval notation for the solutions is \( \{-50, 50\} \).
• Each value within the braces is a solution to the equation provided, where the absolute value of the given expression equates to the specified number.

This notation method is consistent across various applications and quite valuable in both basic and advanced mathematics, ensuring clarity in presenting solutions to both intervals and distinct number sets.