Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)

Verified step by step guidance

1. Start by expanding both sides of the equation to simplify it. Use the distributive property: multiply -2 by each term inside the parentheses on the left side, and -6 by each term inside the parentheses on the right side. This gives you: \(-2 \times x + (-2) \times 3 = -6 \times x + (-6) \times 7\).
2. Rewrite the expanded equation explicitly: \(-2x - 6 = -6x - 42\).
3. Next, collect like terms by adding or subtracting terms to isolate the variable on one side. For example, add \$6x\( to both sides and add \)6$ to both sides to move all variable terms to one side and constants to the other.
4. After simplifying, you will get an equation in the form \(ax = b\), where \(a\) and \(b\) are constants. Analyze this equation: if \(a \neq 0\), then solve for \(x\) by dividing both sides by \(a\); if \(a = 0\) and \(b = 0\), the equation is an identity (true for all \(x\)); if \(a = 0\) and \(b \neq 0\), the equation is a contradiction (no solution).
5. Based on the simplified form, determine whether the original equation is an identity, a conditional equation, or a contradiction, and state the solution set accordingly.

Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

An identity is an equation true for all values of the variable, a conditional equation is true only for specific values, and a contradiction has no solution. Recognizing these types helps determine the nature of the solution set.

Solving Linear Equations

Solving linear equations involves simplifying both sides, distributing constants, combining like terms, and isolating the variable. This process reveals whether the equation holds true universally, conditionally, or not at all.

Solution Sets

The solution set is the collection of all values that satisfy the equation. For identities, it includes all real numbers; for conditional equations, specific values; and for contradictions, it is the empty set.