Problem 12 - Intermediate Algebra

Question

Write each sentence using mathematical symbols. See Examples I through 4 and 6 through 8 . Four times the sum of 5 and \(x\) is not equal to the opposite of 15.

Step-by-Step Solution

Answer

\[4(5 + x) \neq -15\]

Step 1: Identify the Components

First, break down the sentence into parts. Identify 'four times', 'the sum of 5 and x', and 'the opposite of 15'. These parts will form the mathematical expression.

Step 2: Translate 'Four times the sum of 5 and x'

The phrase 'the sum of 5 and x' can be written as the expression \(5 + x\). Multiplying this by four means the expression becomes \(4(5 + x)\).

Step 3: Translate 'Is not equal to the opposite of 15'

The phrase 'the opposite of 15' refers to \(-15\). Therefore, 'is not equal to' translates to \(eq\), resulting in \(eq -15\).

Step 4: Combine the Components into a Mathematical Sentence

Combine the expressions from Steps 2 and 3 to form the inequality: \[4(5 + x) eq -15\]

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics. They allow us to represent numbers and operations in a generalized form.

To understand the structure of an algebraic expression, let's break it down:

• Variables: These are symbols, like \( x \), used to represent unknown values. They can take different values depending on the context.
• Constants: Numbers that have a fixed value, such as 5.
• Coefficients: Numbers that multiply a variable. In the expression \(4(5 + x)\), 4 is the coefficient applying to the entire sum.
• Operations: These are symbols showing mathematical processes, like addition (+) or multiplication (×).

When you read a word problem, it's key to identify these elements and translate them into a concise algebraic expression. This step is crucial because it paves the way for solving real-world problems using math.

Mastering Mathematical Translation

Mathematical translation is a skill that transforms verbal statements into mathematical symbols and expressions. It's like translating sentences into a universal mathematical language.

Let's explore how to translate parts of a sentence into math:

• "The sum of 5 and \(x\)": This indicates you need to add 5 and \(x\), forming the expression \(5 + x\).
• "Four times": This tells you to multiply the preceding expression by 4, resulting in \(4(5 + x)\). Multiplying a sum requires enclosing it in parentheses.
• "The opposite of 15": Here, 'opposite' refers to changing the sign, so this becomes \(-15\).

With these translations, sentences seamlessly become mathematical expressions, ready for further operations or comparisons.

Decoding Inequalities in Mathematics

Inequalities are a way to express that two quantities are not strictly equal. They indicate that something is either larger or smaller than another.

In our problem, we use the "not equal to" symbol \(eq\). This means the expression on one side does not have the same value as the expression on the other side.

• "Is not equal to": This translation results in the notation \(eq\).

In the given example, we created the inequality \(4(5 + x) eq -15\). This inequality tells us that no matter the value of \(x\), the expression \(4(5 + x)\) will not equal \(-15\). Understanding and forming inequalities are crucial in solving many algebraic problems, as they provide a range of possible solutions, illustrating flexibility unlike equations that state exact equalities.