Problem 37 - Elementary and Intermediate Algebra

Question

Solve each equation. If an equation is an identity or a contradiction, so indicate. $$ 0.3(x-4)+0.6=-0.2(x+4)+0.5 x $$

Step-by-Step Solution

Answer

The equation is a contradiction; there is no solution.

Step 1: Distribute the Constants

We start by distributing the constants 0.3 and -0.2 to the terms inside the parentheses.Equation: \[0.3(x-4) + 0.6 = -0.2(x+4) + 0.5x\]After distribution:\[0.3x - 1.2 + 0.6 = -0.2x - 0.8 + 0.5x\]

Step 2: Simplify Each Side

Combine like terms on each side of the equation.For the left side:\[0.3x - 1.2 + 0.6 = 0.3x - 0.6\]For the right side:\[-0.2x - 0.8 + 0.5x = 0.3x - 0.8\]

Step 3: Equate and Solve for x

Set the simplified expressions from each side equal to each other and solve for \(x\).Equation:\[0.3x - 0.6 = 0.3x - 0.8\]Subtract \(0.3x\) from both sides of the equation:\[-0.6 = -0.8\]

Step 4: Determine the Nature of the Equation

Since \(-0.6\) does not equal \(-0.8\), we have an inconsistency. This means the equation is a contradiction, i.e., there is no value of \(x\) that will satisfy the equation.

Key Concepts

Distribution in Algebra

Distribution is a key concept in algebra that allows us to simplify expressions by removing parentheses. This concept relies on the distributive property of multiplication over addition. Whenever you see an expression like \( a(b + c) \), you can transform it to \( ab + ac \). This helps to simplify math problems and make complex expressions more manageable.

In the context of solving equations, distribution is often the first step. We saw this in the exercise where we distributed the constants \( 0.3 \) and \( -0.2 \) across terms inside parentheses.

• The left side, \( 0.3(x-4) + 0.6 \), became \( 0.3x - 1.2 + 0.6 \).
• For the right side, \( -0.2(x+4) + 0.5x \), it simplified to \( -0.2x - 0.8 + 0.5x \).

After this step, you combine like terms to progress further in the equation. Using distribution efficiently makes complex algebraic equations much simpler to handle, setting a clear path forward to solve for unknown variables.

Contradiction in Equations

In algebra, a contradiction occurs when no possible value satisfies the equation. This means that after simplifying and attempting to solve, you end up with a false statement. For example, when you subtract the same term from both sides, but are left with something like \( -0.6 = -0.8 \), you encounter a contradiction. The numbers clearly do not equal each other.

Contradictions often imply that the equation has no solution. Recognizing contradictions helps you understand that certain equations do not have a real-world value for the variable that makes the equation true. In other words, the equation is invalid under any circumstances.

Identifying contradictions early in equations helps save time, as further attempts to solve are unnecessary. It's important to understand this concept because it highlights scenarios where an equation doesn't make sense or cannot exist in practical situations.

Identifying Identities or Contradictions

Identities and contradictions are two special types of equations in algebra. An identity is an equation that is true for every possible value of the variable. For instance, \( x = x \) or \( 2(x + 3) = 2x + 6 \) are identities because they hold true no matter what value \( x \) takes.

Contradictions, as previously discussed, are the opposite—they hold no valid solutions as they result in a false statement, such as \( -0.6 = -0.8 \). Knowing how to differentiate between these two can simplify your problem-solving process.

• Upon solving and simplifying an equation, if you land on something true for all \( x \), it's an identity.
• If it results in a falsehood, it's a contradiction.

Recognizing identities can be equally as useful as identifying contradictions, as it informs whether further calculations are warranted or redundant. When working with equations, always aim to simplify fully and look for these patterns to identify the nature of the solution swiftly.