Problem 71 - Intermediate Algebra

Question

Solve each equation. If the equation is an identity or a contradiction, so indicate. See Example 10. $$ 2 x-6=-2 x+4(x-2) $$

Step-by-Step Solution

Answer

The equation is a contradiction; there are no solutions.

Step 1: Distribute on the Right Side

We begin by distributing the 4 within the expression on the right side of the equation. So, the equation becomes: \[ 2x - 6 = -2x + 4(x - 2) = -2x + 4x - 8 \]

Step 2: Combine Like Terms

Next, we combine the like terms on the right side of the equation. Notice that \(-2x\) and \(+4x\) are like terms. Combine them to get:\[ 2x - 6 = (4x - 2x) - 8 = 2x - 8 \]

Step 3: Simplify the Equation

After combining the terms on the right, the equation now looks like this:\[ 2x - 6 = 2x - 8 \]To simplify the analysis, subtract \(2x\) from both sides:\[ 2x - 6 - 2x = 2x - 8 - 2x \]This simplifies to:\[ -6 = -8 \]

Step 4: Determine the Nature of the Equation

The resulting equation, \(-6 = -8\), is a false statement, which indicates that the original equation is a contradiction. There are no values for \(x\) that can satisfy the equation.

Key Concepts

Equation Solving

Solving equations in algebra is like finding the unknown or mystery number that makes a statement true. The goal is simple: you are given an equation where numbers and variables are balanced, and your job is to determine what numbers those variables represent. Here's how to go about it:

• **Begin by Simplifying**: If there are parentheses, simplify the expressions inside using basic arithmetic.
• **Apply Algebraic Operations**: You can add, subtract, multiply, or divide both sides of the equation to isolate the variable. Keep your equation balanced by performing identical operations on both sides.
• **Aim for Simplicity**: The ideal scenario is to have the variable completely by itself on one side of the equation, matched up with a number on the other.

These steps help you methodically work through an equation. Each adjustment and simplification leads you closer to figuring out that mystery number that will perfectly balance the scales of the equation. With practice, you'll become more effective at anticipating each step!

Distributive Property

In algebra, the distributive property is a handy tool for making calculations easier. It's like spreading a number across the terms inside parentheses. The rule states that for any numbers a, b, and c, the expression \(a(b + c)\) is equal to \(ab + ac\).

Here are simple steps you can follow when using the distributive property:

• **Identify the Multiplier**: Look for a number (or variable with a coefficient) outside the parentheses. This is your multiplier.
• **Distribute to Every Term Inside**: Multiply the number outside by each term inside the parentheses individually.
• **Simplify**: Once you’ve distributed the multiplier, add or subtract the resulting terms as needed to simplify the expression further.

The distributive property helps to transform and simplify equations, especially when you encounter expressions like \(4(x - 2)\). By distributing, you multiply 4 by both \(x\) and -2, turning the expression into \(4x - 8\). This makes it easier to combine like terms and progress in solving the equation.

Contradiction in Algebra

Sometimes, you will encounter equations that just cannot be satisfied; these are known as contradictions. A contradiction occurs when your efforts to solve the equation lead to an impossible or false statement, such as \(-6 = -8\).

Here’s what usually happens:

• **Simplification Leads to Falsehood**: You simplify both sides of an equation until nothing remains unexplained, yet what's left is blatantly false.
• **No Values Satisfy the Equation**: Because the statement is false, there’s no number that can replace the variable to make it true.
• **Identify It as a Contradiction**: Recognizing these false statements is crucial. They tell you that something is off right from the start; there is no solution.

Understanding contradictions helps you to quickly recognize that some equations have no solutions. This knowledge helps to avoid chasing impossible answers and saves you time as you work through algebra problems.