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

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Start by expanding the left-hand side of the equation: $8(x - 3) + 4 = 8x - 21$. Use the distributive property to multiply 8 by both $x$ and $-3$, which gives $8x - 24 + 4$.

Simplify the left-hand side by combining like terms: $8x - 24 + 4$ becomes $8x - 20$.

Rewrite the equation with the simplified left-hand side: $8x - 20 = 8x - 21$.

Next, subtract \$8x$ from both sides to isolate the constants: \(8x - 20 - 8x = 8x - 21 - 8x$, which simplifies to \)-20 = -21$.

Analyze the resulting statement $-20 = -21$. Since this is a false statement, the original equation has no solution and is therefore an inconsistent equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Types of Equations

Equations can be classified as identities, conditional equations, or inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution.

Solving Linear Equations

Solving linear equations involves simplifying both sides, combining like terms, and isolating the variable to find its value. This process helps determine if the equation holds true for all, some, or no values.

Checking Solutions and Equation Validity

After solving, substituting the solution back into the original equation verifies its validity. This step confirms whether the equation is an identity, conditional, or inconsistent based on the truth of the equality.

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