Problem 56 - 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. $$1.5(6 x-3)-7 x=3-(7-x)$$

Step-by-Step Solution

Answer

The equation is a conditional equation with the solution set \( \{0.5\} \).

Step 1: Distribute and Simplify

Start by distributing the term outside the parenthesis on the left side of the equation:\[ 1.5(6x - 3) = 1.5 \times 6x - 1.5 \times 3 = 9x - 4.5 \]The equation becomes:\[ 9x - 4.5 - 7x = 3 - (7 - x) \]

Step 2: Simplify Both Sides

Simplify by combining like terms on both sides:Left side: \[ 9x - 7x - 4.5 = 2x - 4.5 \]Right side: Distribute the negative sign: \[ 3 - 7 + x = x - 4 \]Now the equation is:\[ 2x - 4.5 = x - 4 \]

Step 3: Isolate the Variable

Let's isolate the variable \( x \) by moving all \( x \)-terms to one side and constants to the other side:Subtract \( x \) from both sides:\[ 2x - x - 4.5 = x - x - 4 \]This simplifies to:\[ x - 4.5 = -4 \]Next, add 4.5 to both sides:\[ x - 4.5 + 4.5 = -4 + 4.5 \]Resulting in:\[ x = 0.5 \]

Step 4: Verify the Solution

Substitute \( x = 0.5 \) back into the original equation to verify:Left side: \[ 1.5(6(0.5) - 3) - 7(0.5) = 1.5(3 - 3) - 3.5 = 0 - 3.5 = -3.5 \]Right side: \[ 3 - (7 - 0.5) = 3 - 6.5 = -3.5 \]Both sides are equal, so the solution \( x = 0.5 \) satisfies the equation.

Key Concepts

Solution Set

A solution set is a vital concept in algebra. It consists of values that satisfy the given equation. For the exercise problem, we found that the value that solves the conditional equation is \( x = 0.5 \). This means that when \( x \) is 0.5, both sides of the equation become equal.

Understanding the solution set:

• It helps classify the equation as a contradiction, identity, or conditional.
• A contradiction has no solution. An identity is true for all values, while a conditional equation like our example has one or a few specific solutions.
• Indicating the solution set shows the possible values the variable can take to make the equation true.

Visual aids such as graphs or tables can be useful here. They help show how the equation behaves at different points and confirm where it holds true.

Simplification

Simplification is the process of making an equation easier to work with. Here, we simplified the equation by combining like terms and removing parentheses.

Steps in simplification:

• Distribute any constants across terms in parentheses. For example, \(1.5(6x - 3) \) became \( 9x - 4.5 \).
• Combine like terms on the same side of the equation. This was done by reducing \( 9x - 7x \) to \( 2x \).

Simplification makes equations easier to handle and sets the stage for isolating variables later. It clears any unnecessary complexity and allows for straightforward manipulation.

Variable Isolation

Variable isolation involves rearranging the equation to get the variable on one side alone. The goal is to determine the variable's value that satisfies the equation.

Steps to isolate the variable:

• First, ensure all terms containing the variable are on one side. Here, we moved all \( x \)-terms to one side by subtracting \( x \) from both sides.
• Next, shift constants to the other side to get the variable by itself on one side. We added 4.5 to both sides to isolate \( x \).

This left us with \( x = 0.5 \), which was key in solving our equation. Variable isolation simplifies understanding of the variable's role and the calculation needed.

Equation Verification

Equation verification is the step where you verify that the calculated solution works in the original equation. This ensures the solution is correct and valid.

Process of verification:

• Substitute the solution back into the original equation. We replaced \( x \) with 0.5 in both sides of the equation.
• Simplify both sides to confirm they are equal. In our example, both sides equaled \( -3.5 \), confirming the solution.
• Verification assures that no mistakes were made during simplification or isolation.

It's an important final check in algebra. You confirm that the solution set is accurate for the original equation and ensures confidence in your result.