Problem 83 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Three is not equal to four divided by two.

Step-by-Step Solution

Answer

The inequality is \( 3 \neq 2 \).

Step 1: Identify Key Information

The sentence states, "Three is not equal to four divided by two." Let's identify the mathematical expressions and symbols in this sentence. The number three is a known value, four divided by two is another expression, and the phrase 'not equal to' represents an inequality.

Step 2: Translate Into Mathematical Expression

Start translating the sentence into a mathematical expression. The number three can be represented as the numeral '3'. The expression for four divided by two is \( \frac{4}{2} \). The phrase 'not equal to' can be represented by the inequality symbol \( eq \).

Step 3: Write the Equation or Inequality

Combine the expressions identified in the previous step. The sentence translates into: \[ 3 eq \frac{4}{2} \]

Step 4: Simplify the Expression

Simplify any expressions further if possible. In this case, simplify \( \frac{4}{2} \) to 2. The inequality now becomes: \[ 3 eq 2 \]

Key Concepts

Equations

An equation is a mathematical statement that expresses the equality between two expressions. It usually consists of variables, constants, and arithmetic operations linked by an equality sign (=). The main goal when working with equations is to determine the value of the unknown variable that satisfies the equality.

In the original exercise, we use the symbol 'eq' to demonstrate 'not equal to' instead of just focusing on equations. This concept differs slightly, as it focuses on a comparison rather than finding a specific value. When dealing with equations:

• Identify the expressions on both sides of the equation.
• Use the equality sign to show balance.
• Perform necessary operations to solve for unknown variables.

Understanding the structure of an equation helps in finding solutions and verifying if your final answer satisfies the initial condition set by the equation.

Inequalities

Inequalities represent the relationship between expressions that are not necessarily equal. Instead of an equals sign, inequalities use symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). The key inequality symbol in our exercise is the 'not equal to' (≠) which represents that two expressions do not hold the same value.

When dealing with inequalities, it's important to understand:

• Inequalities do not always have a single solution; they can represent a range of possible values.
• Solutions to inequalities can be expressed as intervals, if they're continuous.
• The sign of the inequality may change when both sides of an inequality are multiplied or divided by a negative number.

In our problem, the inequality is expressed as \( 3 eq 2 \), which simply states that the number three is not equal to two. This shows the fundamental nature of inequalities to depict differences rather than equalities.

Translating Sentences into Math Expressions

Translating sentences into math expressions involves converting written language statements into algebraic expressions or equations using symbols and numbers. This process requires understanding both the vocabulary used in the statement and the mathematical symbols they correspond to.

To successfully translate sentences into math expressions:

• Identify key terms that represent numbers, operations, and relationships (like 'is' for =, 'sum' for +).
• Recognize mathematical symbols that replace words or phrases, such as 'not equal' for ≠.
• Write down the expressions using appropriate variables for unknown quantities.

For the exercise given, the phrase "Three is not equal to four divided by two" translates into the inequality \( 3 eq \frac{4}{2} \). Each component of the sentence directly corresponds to a part of the mathematical expression, highlighting the process of translating everyday language into the structured form of mathematics.