Retaining the Concepts. Solve and determine whether 8(x - 3) + 4 = 8x - 21 is an identity, a conditional equation, or an inconsistent equation.

Verified step by step guidance

1. Start by simplifying both sides of the equation. Distribute the 8 on the left side: 8(x - 3) becomes 8x - 24.
2. Rewrite the equation with the simplified left side: 8x - 24 + 4 = 8x - 21.
3. Combine like terms on the left side: -24 + 4 becomes -20, so the equation is now 8x - 20 = 8x - 21.
4. Subtract 8x from both sides to isolate the constants: 8x - 20 - 8x = 8x - 21 - 8x, which simplifies to -20 = -21.
5. Since -20 does not equal -21, the equation is inconsistent, meaning there are no solutions.

Key Concepts

Identity

An identity is an equation that holds true for all values of the variable involved. For example, the equation 2(x + 1) = 2x + 2 is an identity because it simplifies to the same expression regardless of the value of x. Identifying an equation as an identity means that it is universally valid.

Conditional Equation

A conditional equation is an equation that is true only for specific values of the variable. For instance, the equation x + 2 = 5 is conditional because it is only true when x equals 3. Recognizing an equation as conditional indicates that it has a limited set of solutions.

Inconsistent Equation

An inconsistent equation is one that has no solutions at all. For example, the equation x + 1 = x - 1 is inconsistent because there is no value of x that can satisfy it. Identifying an equation as inconsistent means that it cannot be solved for any real number.