Problem 6 - Intermediate Algebra

Question

Fill in the blanks. The binomials \(x-15\) and \(15-x\) are called ___ because their terms are the same, except that they are opposite in sign.

Step-by-Step Solution

Answer

Opposites.

Step 1: Identify the Binomials

We are given two binomials: \(x - 15\) and \(15 - x\). Our task is to determine the relationship between these binomials.

Step 2: Analyze the Sign Change

Observe that the terms \(x\) and \(15\) are the same in both binomials, but their signs are reversed. In \(x - 15\), \(x\) is positive and \(-15\) is negative, while in \(15 - x\), \(15\) is positive and \(-x\) is negative.

Step 3: Determine the Relationship

When two expressions have the same terms but opposite signs, they are known as 'opposites.' Therefore, \(x - 15\) and \(15 - x\) are opposites.

Key Concepts

Understanding Opposites in Binomial Expressions

In the world of algebra, binomials are expressions that contain two terms, joined by addition or subtraction. Two binomials can be described as 'opposites' if they contain exactly the same terms, but one has the exact reverse signs of the other. For instance, if we take the two binomials \(x - 15\) and \(15 - x\), they contain the same terms: \(x\) and \(15\), just in reverse order. The important detail is that the signs in front of these terms are swapped between the two expressions.

Here is how it works:

• In \(x - 15\), the term \(x\) is positive and \(-15\) is negative.
• In \(15 - x\), the term \(15\) is positive and \(-x\) becomes negative \(x\).

This relationship of having equal terms but opposite signs is what defines them as opposites. Understanding this concept helps in recognizing simplified algebraic forms when solving equations or factoring expressions.

Exploring Sign Change in Algebra

Sign change in algebra occurs when you alter the sign in front of one or more terms in an expression. Recognizing and applying sign changes is crucial in solving many algebraic problems. It's like flipping the sign, either from positive to negative or negative to positive, which can change the value and meaning within an expression.

Consider the binomials \(x - 15\) and \(15 - x\). At first glance, they might look different. However, if you apply a sign change to \(-x + 15\) (which is equivalent to \(15 - x\)), it becomes \(-1(x - 15)\). Here, the entire expression is multiplied by \(-1\), flipping each term's sign in \(x - 15\) to get \(15 - x\).

Recognizing such sign changes can simplify expressions and relationships in algebraic equations. This is particularly useful in solving and factoring, as it reveals equivalent forms of expressions that are not immediately obvious.

Understanding Algebraic Relationships Through Binomials

Algebraic relationships involve expressions and operations that show how mathematical entities are connected. Binomials like \(x - 15\) and \(15 - x\) help us understand these relationships further by exploring what happens when two seemingly different expressions are closely related.

In algebra, understanding that two expressions can represent the same equation despite their initial form is enlightening. Take \(x - 15\) and \(15 - x\). As seen earlier, these expressions are opposites, because the terms are the same but the signs are different.

This demonstrates an important algebraic principle: different expressions can produce the same outcome or solution when used in equations. Recognizing these relationships helps with simplifying, solving, and understanding equations. It allows for flexibility in mathematical manipulations, critical for more advanced algebraic work.