Problem 54 - 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. $$5[1-(3-x)]=3(5 x+2)-7$$

Step-by-Step Solution

Answer

Conditional equation; solution set is \(x = -\frac{9}{10}\).

Step 1: Expand the Equation

We start by expanding the terms inside the brackets on both sides of the equation. For the left side, distribute the 5 across the terms inside the brackets: \[ 5[1-(3-x)] = 5(1 - 3 + x) = 5(-2 + x) = -10 + 5x. \]Now, for the right side: \[ 3(5x + 2) - 7 = 15x + 6 - 7 = 15x - 1. \] This gives us the equation: \[ -10 + 5x = 15x - 1. \]

Step 2: Simplify the Equation

We simplify the equation by combining like terms. Add 10 to both sides:\[ 5x = 15x + 9. \]Next, subtract 15x from both sides to get: \[ -10x = 9. \]

Step 3: Solve for x

To solve for \(x\), divide both sides by -10:\[ x = -\frac{9}{10}. \]

Step 4: Classify the Equation

Examine the solution obtained. Since there is exactly one solution for \(x\), the equation is a conditional equation.

Key Concepts

Contradiction

In mathematics, a contradiction arises when an equation has no possible solution. This contradicts the idea of an equation suggesting it always produces a valid outcome. Consider an example where solving leads to a statement like 0 = 5. This is inherently false, signifying there's no set of values for the variable to satisfy the equation.

Contradictory equations are significant because recognizing them prevents the pursuit of solutions that do not exist. When classifying equations, identifying contradictions saves time, allowing focus on further solvable cases.

Identity Equation

An identity equation is true for every value of the variable used within it. These equations are like a universal truth in mathematics. For instance, if you encountered an equation which simplifies to a statement like 3 = 3 or another tautology, it classifies as an identity.

Identity equations indicate that no matter what number replaces the variable, both sides of the equation will maintain equality. Recognizing these can ease problem-solving as they often involve simplifying expressions or verifying algebraic identities.

Solution Set

The solution set of an equation consists of all possible values that satisfy it. Once an equation is classified, determining its solution set gives clarity to its true nature:

• For a conditional equation, like \(x = -\frac{9}{10}\), the solution set is the specific value that makes the equation true.
• If the equation is an identity, the solution set includes all real numbers, since all values satisfy it.
• Contradictory equations have an empty solution set, as no value makes the equation valid.

Understanding solution sets helps in visualizing the mathematical landscape of equations, making it easier to connect algebraic findings with real-world applications.

Graphical Representation

Graphical representation translates algebraic equations into visual interpretations, assisting in understanding their nature. For example, a graph can highlight where lines representing equations intersect. This intersection is a graphical depiction of the solution set.

When an equation is classified as a conditional, its graph often shows lines intersecting at a single point, indicating a unique solution. For identity equations, the graphical interpretation may appear as overlapping lines, proving equality at all points. Contradiction would show no points of intersection, further illustrating the lack of solutions.

Visual aids like graphs provide intuitive and immediate comprehension of complex algebraic principles, making abstract concepts more tangible.