Problem 128 - Introductory Algebra for College Students

Question

Order the expressions \(|x-y|,|x|-|y|,\) and \(|x+y|\) from least to greatest for \(x=-6\) and \(y=-8\).

Step-by-Step Solution

Answer

The ordered expressions from least to greatest are: |x-y|, |x|-|y|, |x+y|.

Step 1: Substitute the given values

First, substitute x=-6 and y=-8 into the expressions. This gives us: |(-6)-(-8)|, |-6|-|-(-8)|, |(-6)+(-8)|.

Step 2: Evaluate the expressions

Now, simplify each expression into a numerical value: \((|-6|-|-(-8)| = 6-8 = -2)\), \((|(-6)-(-8)| = |2| = 2)\) and \((|(-6)+(-8)| = |-14| = 14)\). Because absolute value is always positive or zero, the absolute value of -2, |-2|, needs to be computed for |-6|-|-(-8)|, resulting in 2.

Step 3: Arrange the values

Now that we have the numerical values of each expression, we can put them in order from the smallest to the largest. This gives us the order: |x-y|, |x|-|y|, |x+y|.

Key Concepts

Absolute Value

The concept of absolute value is essential when working with real numbers, as it represents the distance of a number from zero, regardless of the direction on the number line. Put simply, it converts negative numbers to their positive counterparts and leaves positive numbers and zero unchanged.

In mathematical terms, for any real number \(a\), the absolute value is defined as:

• If \(a \geq 0\), then \(|a| = a\).
• If \(a < 0\), then \(|a| = -a\), which will be a positive number.

This is precisely what we see in the exercise when the absolute values of negative numbers are calculated, for instance, \(|-6| = 6\) and \(|-14| = 14\).

Algebraic Expressions

Algebraic expressions are formations consisting of variables, numbers, and arithmetic operations (addition, subtraction, multiplication, and division). They can be simplified or evaluated but don’t include an equality sign, as equations do. These expressions represent quantities that may vary or be uncertain; hence, we use variables like \(x\) and \(y\) to denote them.

An example from our exercise is \(|x-y|\), which consists of the variables \(x\) and \(y\) and is subject to the operation of subtraction followed by taking the absolute value.

Substitution Method

The substitution method is a technique used in algebra to evaluate expressions or solve equations. It involves replacing variables with their given or known values to simplify an expression or equation. This can help in solving for unknowns or understanding the behavior of the expression.

During the exercise, this method is applied by substituting \(x = -6\) and \(y = -8\) into the various algebraic expressions before proceeding with any further simplifications.

Evaluating Expressions

Evaluating expressions means to calculate the value of an expression for given values of its variables. It requires a systematic approach: use the substitution method to replace the variables with their values, then simplify the expression using the order of operations, which are parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

In our problem, after substituting the values into the absolute value expressions, we evaluated them to find their numerical values, such as \(|-6|-|-8| = 6 - 8 = -2\) which simplifies to \(2\) upon taking the absolute value.