Problem 93 - College Algebra

Question

If \(x\) represents a number, write an English sentence about the number that results in an inconsistent equation.

Step-by-Step Solution

Answer

An example of an English sentence that leads to an inconsistent equation is 'Two times a number, subtracted by itself, equals three.' This results in the inconsistent equation \(2x - x = 3\), which simplifies to \(x = 3\), an equation that is inconsistent because it has no solutions.

Step 1: Understanding Inconsistent Equations

In the realm of algebra, while solving equations, an inconsistent equation is one that having no possible solution. Normally, these occur when two quantities that aren't equal to each other are mistakenly equated. For instance, an equation like \(2 = 3\) is inconsistent because 2 and 3 are different values and cannot be equated.

Step 2: Formulating an English Sentence

The next step is formulating an English sentence that can be translated into an inconsistent mathematics equation. An example of this is: 'Two times a number, subtracted by itself, equals three.'

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a language that allows us to describe patterns, relationships, and changes in quantities. A fundamental aspect of algebra is the formulation of equations, which are statements that show the equality of two expressions.

An equation like \(x + 2 = 5\) is a simple algebraic equation where \(x\) represents an unknown number. To solve for \(x\), one would perform operations that maintain the equality until \(x\) is isolated. However, not all algebraic equations are as straightforward, particularly when they become inconsistent, which leads us to the next concept.

No Solution

An equation with 'no solution' in algebra signifies that there is no value for the variable that would make the equation true. This happens in cases where you arrive at a contradiction during the solving process. A typical example of a contradiction is an inconsistent equation that might look something like \(3x + 4 = 3x + 1\).

Normally, when solving such an equation, both terms containing the variable would cancel out, leaving a statement such as \(4 = 1\), which is clearly false; no number could fulfill this condition. Thus, we declare the equation to be inconsistent, and it is deemed to have 'no solution'. Clearly understanding this concept prevents unnecessary confusion when confronted with non-solvable equations.

Equation Formulation

The process of equation formulation involves translating a verbal description or real-world scenario into an algebraic equation. This skill is an essential component of problem-solving in algebra, as it allows one to frame and tackle a wide variety of tasks.

Illustrating with words

For instance, the statement 'The product of five and a number equals twenty' can be formulated into an equation as \(5x = 20\). When the statement introduces a scenario that is impossible, such as 'A number added to three is less than three', we would end up with an inconsistent equation like \(x + 3 < 3\), which signals no solution.

Moreover, to improve understanding, it is advisable to start with clear and concise sentences when formulating equations. This approach prevents ambiguity and fosters a better grasp of the algebraic representation of the problem at hand.