Problem 82 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Four subtracted from eight is equal to two squared.

Step-by-Step Solution

Answer

The equation is \(8 - 4 = 2^2\).

Step 1: Understand the sentence

The sentence states that 'Four subtracted from eight is equal to two squared.' This means we need to express the operation involving subtraction and square operations as an equation.

Step 2: Translate into mathematical operations

The phrase 'Four subtracted from eight' can be translated to the mathematical operation of subtraction as \(8 - 4\).

Step 3: Represent 'equal to' with an equation sign

The phrase 'is equal to' indicates that we need to place an equation sign \(=\) between the left side and the right side of the equation.

Step 4: Express 'two squared' as an exponentiation

The phrase 'two squared' can be mathematically expressed as \(2^2\).

Step 5: Write the final equation

Combining the results from the previous steps, the equation becomes \(8 - 4 = 2^2\). This represents the sentence as a mathematical equation.

Key Concepts

Expressions and Equations

Understanding the difference between expressions and equations is crucial in math. An expression consists of numbers, variables, and operators, like addition or subtraction, but it doesn't have an equality sign. For example, the phrase 'Four subtracted from eight' can be represented as the expression \(8 - 4\). It shows a calculation but doesn't state a relationship like whether it equals something.

Equations, on the other hand, have an equality sign \(=\). They create a statement about the relationship between two expressions. In our exercise, "Four subtracted from eight is equal to two squared," can be expressed as an equation \(8 - 4 = 2^2\). By setting two sides equal, equations make a claim that can be evaluated and solved. Understanding the transformation from words to equations helps simplify real-world problems into something manageable and solvable in algebra.

Understanding Mathematical Language

Mathematical language can sometimes be perplexing because it involves translating ordinary phrases into precise mathematical symbols. Phrases like 'subtracted from' can imply a reversal in order when written. For instance, 'Four subtracted from eight' translates to \(8 - 4\), not \(4 - 8\).

Similarly, 'equal to' directly tells us to use an equals sign \(=\). Recognizing keywords in math problems helps educators describe quantities and operations accurately.

• 'Squared' means to raise a number to the power of two, like \(2^2\).
• 'Increased by' means addition.
• 'Divided by' means division.

Familiarity with this specific vocabulary makes translating sentences into mathematical equations much more straightforward. Each term in mathematical language has defined meaning and usage, making it easier to solve problems if properly understood.

Basic Algebra Concepts

Algebra often begins with understanding the fundamental concepts of variables, constants, and operations. Variables, like \(x\), represent unknown values that we seek to find. Constants are fixed numerical values, like \(8\) or \(4\).

Another foundational idea is performing operations with numbers and variables using addition, subtraction, multiplication, and division. These operations form the backbone of algebraic expressions and equations. In the provided exercise, the operation used was subtraction, which is one of the four basic operations.

The idea of squaring in math, as depicted in \(2^2\), is also essential. Squaring a number means multiplying the number by itself. It's a basic operation that's frequently used in algebra to represent growth, area calculations, and more.

These fundamental concepts allow us to model numerous real-world situations mathematically, enabling problem-solving and logical reasoning.