Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$\frac{x-1}{2}=\frac{3 x-2}{6}$$

Step-by-Step Solution

Answer

The equation is a contradiction; the solution set is empty, \(\emptyset\).

Step 1: Clear Fractions

First, clear the fractions in the equation by finding a common denominator. The common denominator for 2 and 6 is 6. Multiply each side of the equation by 6 to eliminate the fractions: \[ 6 \times \frac{x-1}{2} = 6 \times \frac{3x-2}{6} \]This simplifies to: \[ 3(x - 1) = 3x - 2 \]

Step 2: Distribute and Simplify

Distribute the multiplication on the left side of the equation: \[ 3(x - 1) = 3 \times x - 3 \times 1 = 3x - 3 \]Now, simplify the equation:\[ 3x - 3 = 3x - 2 \]

Step 3: Isolate Terms

Subtract \(3x\) from both sides to see if the variable terms cancel out:\[ 3x - 3x - 3 = 3x - 3x - 2 \]This leaves:\[ -3 = -2 \]

Step 4: Classify the Equation

The resulting equation \(-3 = -2\) is false since the numbers don't equal one another. This indicates that the original equation is a contradiction, as there are no values of \(x\) that will satisfy the equation.

Key Concepts

Solution Set

In algebra, a "solution set" refers to the set of all possible solutions that satisfy a given equation. When dealing with an equation, our main aim is to find the values of the variable(s) that make the equation true. For the equation \(\frac{x-1}{2}=\frac{3x-2}{6}\), the process leads us to realize that there are no values of \(x\) that will satisfy the equation.
This is because after simplifying and attempting to isolate \(x\), we are left with a false statement \(-3 = -2\).
Since no real number makes this statement true, the solution set is empty. This is a classic result in contradiction scenarios where no solution exists that can balance the equation.

Graphical Representation

Graphical representation can be a helpful tool in algebra to visually understand the solution set of an equation.
For the equation \(\frac{x-1}{2}=\frac{3x-2}{6}\), plotting each side as a separate line on a graph can demonstrate why a solution does not exist.

• The expression \(\frac{x-1}{2}\) can be plotted as a straight line with a particular slope and intercept.
• Similarly, \(\frac{3x-2}{6}\) will plot as another line.

Graphically, a solution to the equation would exist where these two lines intersect each other.
However, if you attempt this with above expressions, you'll find that the lines are parallel.
Parallel lines never intersect, confirming that there's no value of \(x\) where both expressions are equal. This visual aid supports the conclusion that the solution set is indeed empty, reinforcing the algebraic finding of a contradiction.

Clearing Fractions

"Clearing fractions" is a useful technique in algebra that simplifies equations by removing fractions.
This can make it easier to manipulate and solve the equation. When faced with an equation like \(\frac{x-1}{2}=\frac{3x-2}{6}\), starting by clearing the fractions is strategic.
Here’s how it works:

• Identify a common denominator. For the denominators 2 and 6, the least common multiple is 6.
• Multiply each term of the equation by this common denominator to eliminate the fractions.

For this equation, multiplying each side by 6 gives \( 6 \times \frac{x-1}{2} = 6 \times \frac{3x-2}{6} \), which simplifies to \( 3(x - 1) = 3x - 2 \).
The simplification step clears the fractions, turning the equation into a straightforward linear form.
This allows us to explore possible solutions more easily, although in this case, it also leads us to discover the equation is a contradiction with no solution.