Problem 52 - College Algebra

Question

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4(x-7)=4 x+28 $$

Step-by-Step Solution

Answer

The given equation \(4(x-7)=4x+28\) is an inconsistent equation.

Step 1 Simplify Left Side of the Equation

Distribute 4 in the terms inside the bracket on the left side of the equation. This will give \(4 \cdot x - 4 \cdot 7 = 4x - 28\)

Step 2 Compare with Right Side of the Equation

Now both sides of the equation are simplified. Here, compare \(4x-28\) with \(4x+28\). Notice that the only difference lies in the signs. Thus, the given equation is not valid for any real number 'x'.

Step 3 Conclude

Upon comparison, it is observed that the equation could never be validated for any possible values of 'x'. Therefore, it is determined to be an inconsistent equation.

Key Concepts

Identity Equations

An identity equation is an equation that is true for all possible values of the variable involved. These equations describe a fact that is always correct, regardless of the values assigned to the variables. For instance, the equation \( 3(x + 2) = 3x + 6 \) simplifies to \( 3x + 6 = 3x + 6 \), which is true no matter what value 'x' takes on.

A key point to note is that when simplifying both sides of an identity equation, you will end up with the same expression or a universally accepted truth like \( 0 = 0 \). This reaffirms that the equation holds for all values of the variable.

Conditional Equations

Conditional equations are equations that are true only for certain values of the variable. Unlike identity equations, these do not hold for all possible values. For example, \( 2x + 3 = 11 \) is a conditional equation because it is only true when the variable 'x' equals 4.

Solving a conditional equation typically involves isolating the variable on one side of the equation to find the value(s) that makes the equation true. These are the typical kinds of problems one encounters in algebra, as they pose a condition for their solution.

Inconsistent Equations

Inconsistent equations are equations that have no solution. This means that there is no value for the variable that will satisfy the equation. A classic example is the equation \( x + 5 = x - 3 \), which upon simplification implies \( 5 = -3 \), a false statement.

As encountered in the original exercise, comparing the left side \( 4x - 28 \) with the right side \( 4x + 28 \) after simplification reveals a contradictory equation. As such, this equation is inconsistent because it's impossible to find a real number 'x' that will satisfy both sides of the equation simultaneously.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves rewriting expressions in their simplest form. This process usually entails combining like terms, distributing multiplicative factors, and performing arithmetic operations where applicable.

To simplify effectively, one must understand the order of operations and the properties of real numbers. For instance, in the exercise provided, simplifying the left side involves distributing the 4 into the terms within the brackets. This involves applying the Distributive Property of multiplication over addition. Always check both sides of an equation after simplifying to ensure their validity, which can lead to identifying whether an equation is an identity, conditional, or inconsistent.