Problem 75 - College Algebra Essentials

Question

Combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. \(\frac{2}{x}+\frac{1}{2}=\frac{3}{4}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\) and it is a conditional equation.

Step 1: Getting rid of Denominators

Multiplying each term by the least common denominator (LCD) of 4x will eliminate the denominators from the equation. As such, the equation becomes: \(4x*\frac{2}{x} + 4x*\frac{1}{2} = 4x*\frac{3}{4}\). This simplifies to \(8 + 2x = 3x\)

Step 2: Solving for x

Use algebraic manipulation to solve for \(x\). First, subtract \(2x\) from both sides of the equation, resulting in \(8 = x\).

Step 3: Identify the Type of Equation

Substitute \(8\) for \(x\) in the original equation and check if both sides are equal. If both sides equal, then the equation is a conditional equation, as it has exactly one solution. If they don't equal, then the equation is an inconsistent equation. However, as both sides are equal when \(x=8\), we classify it as a conditional equation.

Key Concepts

Conditional Equations

A conditional equation is a type of equation that is true for certain values of the variable but not for others. In our exercise, we found that the equation \(\frac{2}{x} + \frac{1}{2} = \frac{3}{4}\) simplifies to \(8 + 2x = 3x\). Solving gives \(x = 8\) as the only solution.

When you solve a conditional equation, you find a specific value (or values) that make the equation true. If you substitute any other number in place of \(x\), the equation will not hold.

Here are some characteristics of conditional equations:

• They have one or a few specific solutions.
• If the variable is substituted with a wrong value, the equation does not balance.
• They are common in algebraic problem-solving.

When solving, always check your solution by substituting it back into the original equation to confirm it balances.

Inconsistent Equations

Inconsistent equations are those that have no solution. This means there are no values for which the equation is true. These typically occur when, through manipulation, you reach a contradiction—like 0 = 5—which is obviously false.

When dealing with algebra, identifying an inconsistent equation is crucial because it prevents futile attempts to find a solution.

Key features of inconsistent equations include:

• They result in a false statement after simplifying.
• No value of the variable makes both sides of the equation equal.

Handling inconsistent equations can involve recognizing contradictions early, saving time and effort. Always verify the structure of your equation to rule out this possibility.

Identity Equations

An identity equation is an equation that is true for all values of the variables involved. Unlike conditional equations, identity equations hold irrespective of what values are used in them.

A classic example is \(x + 0 = x\); it's always true, no matter what \(x\) represents.

Features of identity equations include:

• They simplify to a statement that is always true, such as 0 = 0.
• They have an infinite number of solutions.
• You can't pin down a specific solution like you would in a conditional equation because every number works.

Understanding these can help students recognize generalized truths in algebra and distinguish them from other equation types. Always simplify fully to identify identities.