Problem 86 - Algebra 1

Question

Is the opposite of the absolute value of a number =ver the same as the absolute value of the opposite of the number? In other-words, is it ever true that \(-|x|=|-x| ?\) Explain.

Step-by-Step Solution

Answer

No, it is not true unless 'x' is zero. If 'x' is negative, \( -|x| \) will be positive while \( |-x| \) will stay positive. Likewise, if 'x' is positive, \( -|x| \) will be negative while \( |-x| \) will stay positive. When 'x' is zero, both expressions equal zero, so they are equal in that case.

Step 1: Understanding Absolute Value

To start, it's important to understand what an absolute value is. The absolute value of a number 'n' (denoted as |n|) always yields a nonnegative value. If 'n' is negative, |-n| is positive. If 'n' is positive, |n| is also positive.

Step 2: Work Out \( -|x| \)

The expression \( -|x| \) is the negative of the absolute value of 'x'. Regardless of whether 'x' is positive or negative, |x| will be nonnegative, hence -|x| will always be nonnegative or zero (if 'x' is zero).

Step 3: Work Out \( |-x| \)

On the other hand, the expression \( |-x| \) is the absolute value of the negative of 'x'. Similarly, regardless of whether 'x' is positive or negative, |-x| will always be nonnegative or zero (if 'x' is zero).

Step 4: Comparing \( -|x| \) and \( |-x| \)

Comparing the two expressions, we can see a discrepancy. While \( -|x| \) is always nonpositive or zero, \( |-x| \) is always nonnegative or zero. This means that these two expressions are not equivalent, except for when 'x' equals zero.

Key Concepts

Absolute Value Properties

When tackling algebraic problems, understanding absolute value properties is crucial. An absolute value, symbolized as \( |x| \), represents the distance of a number \( x \) from zero on the number line, without considering its direction. The key property to remember is that the absolute value of any real number is always nonnegative (

• \( |3| = 3 \)
• \( |-3| = 3 \)

). This property comes in handy when comparing expressions involving absolute values. One interesting characteristic to also acknowledge is the effect of applying a negative sign outside the absolute value symbol, \( -|x| \), which mirrors the positive result to the negative side of the number line.

Comparing \( -|x| \) with \( |-x| \), we notice a common mistake. Students might initially think these are the same because they seem to involve similar manipulations of \( x \). However, by leveraging the absolute value properties, we understand that the negative outside changes the sign while the internal negative does not change the magnitude. Recognizing these properties helps avoid confusions and errors in computing and comparing expressions with absolute values.

Negative Number Concepts

Delving into the realm of negative numbers can sometimes be intimidating, but mastering this concept is a critical step in understanding algebra. A negative number is simply a real number that is less than zero. These numbers are essential for representing values like debt or temperature below zero.

It is important to distinguish between the number itself and its negative counterpart, which is denoted by a minus sign (e.g., \( -5 \) is the negative of \( 5 \)). When dealing with negative numbers, remember that multiplying or dividing two negative numbers results in a positive product or quotient, reflecting the idea that a double negative results in a positive (

• \( -(-5) = 5 \)
• \( -2 \times -3 = 6 \)

).

This concept plays a vital role when working with absolute values, where misconceptions may arise. For instance, the absolute value of a negative number is the number without its negative sign, embodying the distance from zero without the direction. By clarifying these negative number concepts, we can correctly interpret and solve problems involving absolute values and negatives, such as in the exercise comparing \( -|x| \) and \( |-x| \).

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and form a combination of numbers, variables (like \( x \) or \( y \)), and arithmetic operations (such as addition, subtraction, multiplication, division). A simple example of an algebraic expression is \( 3x + 2 \), where \( x \) represents an unknown value, and the numbers \( 3 \) and \( 2 \) are coefficients and constants, respectively.

In the context of our example, \( -|x| \) and \( |-x| \) are algebraic expressions involving the variable \( x \), the absolute value operation, and negation. Understanding how to manipulate these expressions according to algebraic rules and properties is essential. For instance, when solving for \( x \) in equations involving absolute values, it is necessary to consider both positive and negative solutions since the absolute value operation removes any negative sign.

Correctly interpreting and manipulating such expressions is imperative for solving algebraic problems. By practicing different expressions, students can reinforce their ability to solve more complex equations and deepen their comprehension of algebraic concepts.