Question

Evaluate the expression. $$ -|13-12.1| $$

Short answer

The solution of the expression \(-|13 - 12.1|\) is \(-0.9\).

Step-by-step solution

Evaluate the expression inside the absolute value

Subtract 12.1 from 13. Therefore, \(13 - 12.1 = 0.9\).

Apply the absolute value

The absolute value of 0.9 is 0.9 as it is a positive number.

Apply the negative sign

Finally, negate the absolute value. Therefore, the value of the expression is \(-0.9\).

Key concepts

Evaluate Expressions

Evaluating algebraic expressions is a foundational skill in mathematics that involves substituting numbers for variables and performing operations according to the established rules of arithmetic. When approaching problems that require you to evaluate expressions, particularly with absolute values and negative numbers, it's important to tackle them step by step.

First, assess the expression to identify which parts need to be computed first by following the order of operations (PEMDAS/BODMAS). Next, if the expression includes an absolute value, as in the given exercise \( -|13-12.1| \), interpret the absolute value as the distance a number is from zero on a number line, regardless of the direction. After simplifying the expression inside the absolute value, apply the absolute value rules before performing any additional operations, such as applying a negative sign.

Absolute Value Properties

The absolute value of a number can be thought of as its distance from zero without considering direction. It's represented by two vertical bars surrounding the number or expression, such as \( |x| \). There are key properties of absolute values that are crucial to understand:

• Non-negativity: Absolute value is always non-negative, as distance cannot be negative.
• Positive Numbers: If the number is positive, including zero, the absolute value is the same as the number: \( |x| = x \), for \( x \geq 0 \).
• Negative Numbers: For a negative number, the absolute value is the number's opposite: \( |x| = -x \), for \( x < 0 \).
• Zero: The absolute value of zero is zero: \( |0| = 0 \).

When evaluating expressions, remember to use these properties accordingly after simplifying any expressions within the absolute value. In the given exercise, the expression within the absolute value is a positive number, hence the absolute value doesn't change its sign.

Negative Numbers

Understanding negative numbers is essential when working with real-life applications and mathematical concepts. A negative number is a number that is less than zero, and it represents the absence of a quantity or being below zero. In the context of the absolute value, a negative number outside of the absolute value symbols, such as in our example \( -|13-12.1| \), changes the sign of the value that's within the absolute value.

When handling negative numbers, keep in mind that

• Multiplying or dividing two negative numbers gives a positive result.
• Adding a negative number is the same as subtracting its absolute value.
• Subtracting a negative number is the same as adding its absolute value.

After calculating the absolute value in our example, we apply the negative sign, resulting in \( -0.9 \) as the final answer. Understanding these concepts ensures precision in evaluating expressions with mixed positive and negative values, especially within the structure of an absolute value expression.