Problem 39 - Algebra 2

Question

REVIEW If \(x\) is a real number, for what values of \(x\) is the equation \(\frac{4 x-16}{4}=x-4\) true? \(\mathbf{F}\) all values of \(x\) G some values of \(x\) H no values of \(x\) J impossible to determine

Step-by-Step Solution

Answer

The equation is true for all values of \(x\).

Step 1: Simplify the Left Side of the Equation

The equation is given by \( \frac{4x - 16}{4} = x - 4 \). Start by simplifying the left-hand side. Divide both terms of the numerator by 4: \[ \frac{4x - 16}{4} = \frac{4x}{4} - \frac{16}{4} = x - 4. \]

Step 2: Recognize Simplified Equation

After simplifying the left-hand side, the equation reduces to: \[ x - 4 = x - 4. \] This is a tautology because both sides are exactly the same.

Step 3: Determine the Truth of the Equation

Since \( x - 4 = x - 4 \) is a tautology, it holds true for all real numbers \( x \). Therefore, no matter what value is substituted for \( x \), both sides will always be equal.

Key Concepts

Understanding Tautology

In the realm of mathematics, a tautology is an equation or expression that is true in every possible interpretation. In simpler terms, it's always true. A classic example of a tautology is when both sides of an equation are identical. This means no matter what value you substitute into the equation, it remains valid.

For instance, the equation \( x - 4 = x - 4 \) is a tautology. Both sides of the equation are identical, resulting in a statement that is universally true. It's like saying "it is what it is" – there's nothing to solve because it's always accurate.

In practical situations, recognizing a tautology can be very helpful. It tells us the equation holds for every possible value of the variable involved. In the given exercise, since the equation simplifies to a tautology, we can conclude that the statement "for all values of \( x \)" is correct. This understanding is key in determining the solution to problems involving real numbers.

Equation Simplification

Equation simplification is a fundamental technique in solving mathematical problems. It's the process of reducing an equation to its simplest form. This often involves reducing expressions and making them easier to work with.

Consider the initial equation \( \frac{4x - 16}{4} = x - 4 \). To simplify, we divide each term in the numerator by 4, giving us \( x - 4 \). This step is crucial because it allows us to see the equation more clearly.

The goal is to reduce complexity while preserving the equation's integrity. Simplifying can involve:

• Combining like terms
• Reducing fractions
• Distributing factors

These techniques help us to understand better and solve equations, revealing solutions that might not be immediately obvious before simplification. In this case, simplification revealed the tautological nature of the equation.

Exploring Algebraic Expressions

Algebraic expressions form the backbone of algebra and mathematics generally. They consist of variables, constants, and operators like addition or subtraction. Understanding how to manipulate these expressions is critical for solving algebraic equations.

For instance, the expression \( 4x - 16 \) from the problem can be broken down into terms, which are parts of the expression that are added or subtracted. Here, \( 4x \) is a term, and \( -16 \) is another. Manipulating algebraic expressions often involves operations such as:

• Factoring
• Expanding
• Simplifying

These operations make it easier to solve equations and understand their behavior. By learning to handle algebraic expressions effectively, students can better navigate complex problems and find solutions confidently. This grasp of algebraic expressions, coupled with simplification techniques, allowed us to identify the tautology in the original exercise.