Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 1-6, determine whether the equation is an identity or a conditional equation. $$ 2(x-1)=2 x-2 $$

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Simplify LHS and RHS

The Left-hand side (LHS) of the equation is \(2(x-1)\). Distributed property of multiplication over addition/subtraction can be applied here which gives \(2x-2\). The right-hand side (RHS) is \(2x-2\). So, after simplification, the equation becomes \(2x-2 = 2x-2\).

Step 2: Compare LHS and RHS

Now, compare the simplified form of the equation. If both sides of the equation match for all x, then it's an identity. Here, \(2x-2\) and \(2x-2\) are the same and equal for all real values of x.

Step 3: Declare the type of the equation

Since both sides of the equation are equal for every x, we declare the equation to be an identity.

Key Concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It consists of two expressions connected by an equal sign "=". For example, in the equation \(2(x-1) = 2x - 2\), each side of the equation, known as the Left-hand Side (LHS) and Right-hand Side (RHS), must be balanced.

• LHS: The expression on the left of the equal sign. Here, it's \(2(x-1)\).
• RHS: The expression on the right of the equal sign. Here, it's \(2x - 2\).

To solve an equation means to find the value(s) of the variable(s) that make the equality true. The primary goal is to isolate the variable on one side of the equation to determine its value. This involves simplifying expressions and using various algebraic rules. Keep in mind that equations can be of different types and may have one, more, or no solutions at all.

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by each term within a parenthesis. It makes simplifying expressions more manageable.For example, using the distributive property on \(2(x-1)\) means:

• Multiply \(2\) by \(x\): This gives \(2x\).
• Multiply \(2\) by \(-1\): This gives \(-2\).

Putting it together: \[2(x-1) = 2x - 2\] This property is handy when you are dealing with complex expressions, as it helps break them down into simpler forms. Understanding how to apply the distributive property allows you to manipulate equations more effectively, especially when simplifying both LHS and RHS to see if they can lead to a simpler, equivalent form.

Conditional Equations

Conditional equations are equations that are true only under specific conditions or for certain values of the variables involved. Unlike identities, which are true for all values of the variable, conditional equations might only be true in specific instances. For example, the equation \(x + 2 = 5\) is conditional because it only holds true when \(x = 3\).In our original exercise, however, after simplifying both LHS and RHS, the equation \(2x - 2 = 2x - 2\) became an identity because both sides are equal for any real number \(x\). To distinguish if an equation is conditional, you need to compare simplified LHS and RHS:

• If they match for only specific values of \(x\), it’s conditional.
• If they match for every value, the equation becomes an identity.

This concept is important in algebra as it helps categorize and solve different types of equations more effectively.