Problem 72 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$-2-|-2|$$

Step-by-Step Solution

Answer

Answer: -4

Step 1: Evaluate the Absolute Value

First, we must evaluate the absolute value expression. The definition of absolute value is: $$|x| = \begin{cases} x, & \text{if} \ x>=0\\ -x, & \text{if} \ x<0 \end{cases}$$ In this case, our value within the absolute value expression is -2, which is less than 0. Therefore, we would use the second condition, which is: $$|-2| = -(-2)$$ Now, we evaluate the expression: $$|-2| = 2$$

Step 2: Complete the Arithmetic Operation

Now that we have the value of the absolute value expression, we can substitute it back into the original expression: $$-2-|-2| = -2-2$$ Perform the subtraction: $$-2-2 = -4$$ The simplified expression without absolute values is: $$-4$$

Key Concepts

Absolute Value Definition

Understanding the absolute value is crucial for mastering precalculus and various branches of mathematics. It is defined as the distance a number is from zero on a number line, without considering which direction from zero the number lies. The absolute value of a number is always non-negative.

For any real number \( x \), the absolute value is written as \( |x| \) and is defined by the piecewise function:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \), which converts a negative number to its positive counterpart.

For example, both \( |5| \) and \( |-5| \) equal 5 since both 5 and -5 are five units away from zero on the number line.

Arithmetic Operations

Arithmetic operations are the foundation of mathematics, encompassing addition, subtraction, multiplication, and division. These operations are used to compute quantities, compare values, and understand relationships between numbers.

When simplifying expressions that involve absolute values, it's essential first to evaluate the absolute values before proceeding with other arithmetic operations. In the given exercise, after evaluating the absolute value \( |-2| \) as 2, we perform the subtraction \( -2 - 2 \), resulting in -4. By understanding the correct order in which to execute these operations, students can accurately simplify complex expressions.

Precalculus Mathematics

Precalculus is a course that prepares students for calculus, and it's where they encounter fundamental topics like absolute values, functions, complex numbers, and trigonometry. One integral skill in precalculus is the ability to simplify expressions, which is a stepping stone for solving more complicated problems.

While working with absolute values, students learn the importance of piecewise-defined functions and how they are applicable in a variety of contexts. Simplifying absolute value expressions often includes a mix of other arithmetic operations, and mastering this skill builds a strong foundation for future topics in calculus and higher-level mathematics.

Understanding and applying these concepts correctly are vital for any student aiming to progress in their mathematical education. As seen in the exercise, simplifying an absolute value expression requires careful attention to definitions and the order of operations to arrive at the correct solution.