Problem 62 - Elementary Algebra

Question

Simplify. $$ ||-5|-| 10 \mid $$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Evaluate Single Absolute Values

First, evaluate the inner absolute values: - The absolute value of \(-5\) is \(| -5 | = 5\) because absolute value transforms negative numbers into their positive counterparts.- The absolute value of \(10\) is \(| 10 | = 10\) since it is already positive.

Step 2: Calculate the Expression Inside the Outer Absolute Value

Next, substitute the evaluated values into the expression inside the outermost absolute value:\[|-5| - | 10 | = 5 - 10 \]Proceed with the calculation to find the result.\[5 - 10 = -5\]

Step 3: Evaluate the Outer Absolute Value

Finally, evaluate the absolute value of the result from Step 2:The absolute value of \(-5\) is \(| -5 | = 5\), as absolute value converts negative numbers to positive numbers.

Key Concepts

Simplifying Expressions with Absolute Values

Simplifying expressions in algebra requires us to break down complex components into simpler ones. Absolute value expressions, like \(||-5|-| 10|\), involve calculating the distance of a number from zero on the number line. To simplify:

• First, find the absolute values individually. For \(-5\), this becomes \(5\). For \(10\), it's already \(10\).
• Then, perform the operation inside the absolute value brackets: \(5 - 10 = -5\).
• Finally, apply absolute value again: \(|-5| = 5\).

This step-by-step breakdown ensures you deal correctly with inner and outer layers of absolute values.

Algebra Fundamentals: Understanding Absolute Value

Absolute value is a fundamental concept in algebra that measures how far a number is from zero, regardless of direction. The symbol \(|x|\) indicates this measurement.

• Positive numbers remain unchanged when taking absolute value.
• Negative numbers are converted to their positive form.

For example, \(|-5|\) becomes \(5\) and \(|10|\) stays \(10\). Remembering this rule helps prevent mistakes when simplifying expressions that involve absolute values.

Essential Mathematical Operations in Expressions

When simplifying expressions, it's crucial to understand the order of operations, especially when dealing with absolute values.

• Always resolve expressions within absolute value brackets first.
• Next, perform subtraction, addition, or other operations within the expression.
• Finally, re-evaluate the absolute value if necessary.

In our exercise, this meant calculating the inner values first (\(|-5|\) and \(|10|\)), followed by the subtraction, and then applying absolute value one more time if needed. This systematic approach ensures accuracy and clarity in problem-solving.