Problem 55 - 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. $$7 x-3[5 x-(5+x)]=1-4 x$$

Step-by-Step Solution

Answer

The equation is conditional with the solution set \(\{14\}\).

Step 1: Distribute inside the brackets

Start by distributing the negative sign inside the brackets in the equation: \(7x - 3[5x - (5 + x)] = 1 - 4x\). First, remove the inner parentheses: \(5x - (5 + x)\) becomes \(5x - 5 - x\).

Step 2: Simplify the expression in brackets

Now simplify the expression inside the brackets: \(5x - x - 5\), which simplifies to \(4x - 5\).

Step 3: Distribute the 3 across the simplified bracket expression

Multiply \(-3\) with each term inside the bracket: \(-3(4x - 5)\), which gives \(-12x + 15\).

Step 4: Simplify the original equation

Substitute the distributed expression back into the original equation: \(7x - 12x + 15 = 1 - 4x\).

Step 5: Combine like terms

Combine the \(x\) terms on the left side of the equation: \(7x - 12x\) simplifies to \(-5x\). This results in the equation \(-5x + 15 = 1 - 4x\).

Step 6: Isolate x

Add \(4x\) to both sides of the equation to isolate \(x\) terms on one side: \(-5x + 4x + 15 = 1\), which simplifies to \(-x + 15 = 1\).

Step 7: Solve for x

Subtract 15 from both sides to find \(x\): \(-x + 15 - 15 = 1 - 15\), giving \(-x = -14\). Hence, \(x = 14\).

Step 8: Classify the equation and determine the solution set

Since we arrived at a specific solution \(x = 14\), this is a conditional equation. The solution set is \(\{14\}\).

Key Concepts

Equation Classification

When dealing with equations, it's essential to identify which type we are working with, as it tells us about the nature of the solutions. There are three primary types of equations:

• Contradiction: An equation that is never true. For example, equations like \( x + 2 = x - 3 \) have no solutions.
• Identity: An equation that is always true for any value of the variable. An example is \( x + 1 = x + 1 \).
• Conditional Equation: This type of equation is true only for particular values of the variable. For example, the equation \( 2x = 4 \) has a solution \( x = 2 \).

Identifying the type of equation helps us understand if we should expect one, none, or infinite solutions. In our example, we are working with a conditional equation because we find a specific value \( x = 14 \) that satisfies it.

Solution Set

The solution set of an equation includes all values that make the equation true. It's a collection of numbers that solve the problem. In math, we often use curly braces \( \{ \} \) to show the collection.
For the given conditional equation, after carefully simplifying and solving the equation, we found that the solution is \( x = 14 \). This means our solution set is \( \{14\} \).

• Sometimes, a solution set can have more than one value. For example, \( x^2 = 4 \) has a solution set of \( \{-2, 2\} \).
• If an equation is a contradiction, the solution set is empty, represented as \( \{ \} \).

Understanding solution sets allows you to see clearly what answers are valid within the context of a particular equation.

Distribution in Algebra

Distribution in algebra involves applying the distributive property. This helps us simplify expressions by eliminating parentheses:

• Distributive Property Formula: \( a(b+c) = ab + ac \).
• Use this property to break down expressions within parentheses.

In our solved example, initially, we had to distribute the negative sign and then the number 3 through the terms inside the brackets: \(-3(4x - 5)\) became \(-12x + 15\).
Being able to distribute correctly simplifies expressions and is a crucial step in solving complex equations. This method allows us to transform complicated terms into simpler ones, which are easier to manage and solve.

Simplifying Equations

Simplifying equations involves reducing them to their simplest forms so that solutions become easier to find. The key steps to simplify an equation are:

• Combine like terms: Bring similar terms together. E.g., \( 7x - 12x \) simplifies to \(-5x \).
• Isolate the variable: Use algebraic operations to get the variable alone on one side of the equation.
• Perform arithmetic operations: Balance each side of the equation by adding, subtracting, multiplying, or dividing.

For our exercise, the expression \( 7x - 12x + 15 = 1 - 4x \) is simplified by combining and rearranging terms to eventually solve for \( x \). Each simplification step helps clarify what we're working with and sets the stage for finding the solution efficiently.