Determine whether each equation is an identity, a conditional equation, or a contradic-tion. Give the solution set. -6(2x+1) - 3(x-4) = -15x+1

Verified step by step guidance

1. Distribute the -6 across the terms inside the parentheses: \(-6(2x + 1) = -12x - 6\).
2. Distribute the -3 across the terms inside the parentheses: \(-3(x - 4) = -3x + 12\).
3. Combine like terms on the left side of the equation: \(-12x - 6 - 3x + 12 = -15x + 6\).
4. Set the simplified left side equal to the right side: \(-15x + 6 = -15x + 1\).
5. Compare the coefficients and constants on both sides to determine if the equation is an identity, conditional, or contradiction.

Key Concepts

Identity

An identity is an equation that holds true for all values of the variable involved. This means that no matter what value you substitute for the variable, both sides of the equation will always be equal. For example, the equation 2(x + 1) = 2x + 2 is an identity because it simplifies to the same expression on both sides.

Conditional Equation

A conditional equation is an equation that is true only for specific values of the variable. These equations typically have a finite number of solutions. For instance, the equation x + 2 = 5 is conditional because it is only true when x equals 3.

Contradiction

A contradiction is an equation that has no solutions at all, meaning there are no values for the variable that can make the equation true. An example of a contradiction is the equation x + 1 = x, which simplifies to 1 = 0, a statement that is always false.