Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the equation is an identity or a conditional equation. $$ 2(x+1)=2 x+1 $$

Step-by-Step Solution

Answer

The equation \(2(x+1) = 2x + 1\) is neither an identity nor a conditional equation as it is not true for any value of 'x' nor for all values of 'x'.

Step 1: Distribute Multiplication Over Addition

First, implement the distributive property on the left-hand side of the equation. Therefore, \(2(x+1)\) becomes \(2x+2\).

Step 2: Simplify the Equation

After distributing, the simplified equation is \(2x + 2 = 2x + 1\).

Step 3: Check for Identity

For an equation to be an identity, both sides of the equation must be the same for all x. Here, \(2x + 2\) is not equal to \(2x + 1\) for all x, so this is not an identity.

Step 4: Solve for x

First, subtract \(2x\) from both sides of the equation to isolate x on one side. This leaves us with an equation of \(2 = 1\), which is a false statement. Therefore, there are no solutions for x, and the original equation is not a conditional equation.

Key Concepts

Distributive Property

Understanding the distributive property is crucial when dealing with algebraic expressions. It involves the multiplication of a single term by two or more terms within a parenthesis. In essence, it 'distributes' the multiplication over addition or subtraction. When you encounter an expression like (2(x + 1)), the distributive property is applied by multiplying the 2 to both (x) and (1), which results in 2x + 2. This property is invaluable for simplifying equations and setting the stage for solving them.

When applying the distributive property, always remember to multiply each term inside the parenthesis by the term outside. This often involves multiplying variables, numbers, or even more complex expressions. Equational balance is maintained, which is a cornerstone principle in algebra.

Simplifying Equations

The process of simplifying equations reduces them to their most basic form, making them easier to solve. Simplification may involve combining like terms, using the distributive property, and canceling out terms. For the given exercise, after employing the distributive property to the equation (2(x + 1)), we get (2x + 2). Simplification doesn't change the equation's value but makes the subsequent steps in solving it more apparent.

Simplification can also include removing parentheses, merging constants, and reducing fractions to their lowest terms. Each of these steps leads to an equation that is easier to interpret and solve. Remember that any operation done to one side must also be done to the other to maintain equality, which reflects the concept of a balanced equation.

Solving Equations

Solving equations is the process of finding the value(s) of the variable(s) that satisfy the equation. The goal is to isolate the variable on one side of the equation, making it clear what the variable equals. This process can involve several steps, including distributing, combining like terms, adding or subtracting terms on both sides, and finally dividing or multiplying to solve for the variable.

In our exercise, we attempt to solve the equation after simplifying it to 2x + 2 = 2x + 1. Upon subtracting 2x from both sides, we aim to isolate the variable x, but we encounter 2 = 1, an impossibility which tells us there are no valid solutions. This points out that not every equation will have a solution; some lead to false statements indicating no possible values for x satisfying the original equation.