Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. -0.6(x-5)+0.8(x-6) = 0.2x - 1.8

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First, distribute the constants on the left side of the equation to remove the parentheses: apply the distributive property to both terms, so calculate $-0.6(x - 5)$ and $0.8(x - 6)$ separately.

After distributing, combine like terms on the left side to simplify the expression into the form $ax + b$.

Rewrite the equation with the simplified left side and the right side as given: $ax + b = 0.2x - 1.8$.

Next, get all variable terms on one side and constant terms on the other side by subtracting \$0.2x$ from both sides and adding or subtracting constants accordingly.

Finally, analyze the resulting equation: if the variable terms cancel out and you get a true statement (like $c = c$), it's an identity; if you get a false statement (like $c = d$ where $c \neq d$), it's a contradiction; otherwise, solve for $x$ to find the solution set for a conditional equation.

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주요 개념

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Types of Equations: Identity, Conditional, and Contradiction

An identity is an equation true for all values of the variable, a conditional equation is true 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 expressions, combining like terms, and isolating the variable. This process helps find the values that satisfy the equation or determine if no or all values work.

Solution Set of an Equation

The solution set is the collection of all values that make the equation true. It can be a single value, multiple values, all real numbers, or empty, depending on the equation's type.