Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. An equation that is false for all replacement values for the variable is called a _____ .

Step-by-Step Solution

Answer

An equation that is false for all replacement values for the variable is called a contradiction.

Step 1: Understanding the Problem

We need to fill in the blank with a term that describes an equation that is false for any value assigned to its variable(s). This means no matter what value is used, the equation never holds true.

Step 2: Definition Recall

Recall the definition of different types of equations: - **Identity**: True for all values of the variable. - **Conditional equation**: True for particular values of the variable. - **Contradiction**: False for all values of the variable.

Step 3: Identifying the Correct Term

Since the equation is false for all possible values of the variable, it fits the definition of a 'contradiction'. A contradiction in mathematics is an equation that has no solutions under any circumstances.

Key Concepts

Contradiction

A contradictory equation is quite unique because it's inherently false, regardless of which number you substitute for the variable. Imagine an equation that, no matter what number or value you think of, just doesn't fit or fulfill it. Such equations hold no truth under any circumstances. For example, the equation \( x + 2 = x + 3 \) is a contradiction. - No solution exists that will make this equation true. - With any number substituted for \(x\), the equation remains untrue. These contradictions often occur when you simplify an equation and end up with a non-sensical or impossible statement, like stating that 1 equals 2.

Identity

An identity is an equation that holds an inherent truth for any value the variable takes. Picture it like a universal truth in mathematics. An identity tells us that both sides of the equation are always equal no matter what you plug into the variable. Examples of identity equations are:- \( 3(x + 4) = 3x + 12 \)- \( x + 0 = x \)Whenever you substitute any real number for \(x\) in these equations, the outcome will always be balanced and correct. This characteristic makes identities very useful in algebra and other branches of mathematics when proving other theorems or simplifying expressions.

Conditional Equation

Conditional equations are a little different from contradictions and identities. They stand as true only when specific values satisfy them. They're not always false or always true like the others, but instead depend on particular numbers. Consider the equation \( 2x + 3 = 7 \):- This equation is only true when \(x = 2\). - Other values for \(x\) will make the equation false.These kinds of equations typically appear in problems where you're asked to solve for a variable, providing valuable practice in understanding how to isolate and identify solutions. Conditional equations help in developing problem-solving skills and improving mathematical reasoning.