Solve for x: $2(2x + 4) = 4x + 8$.
The given equation is \(2(2x + 4) = 4x + 8\). We are asked to find the value of \(x\) that satisfies this linear equation. The equation involves a single variable \(x\), which is assumed to be a real number. Our objective is to determine whether there is a unique solution, no solution, or if the equation holds for all real values of \(x\).
We begin by distributing the constant factor 2 into the expression inside the parentheses on the left-hand side of the equation:
\[2(2x + 4) = 2 \cdot (2x) + 2 \cdot (4) = 4x + 8.\]
Now, we substitute this back into the original equation:
\[4x + 8 = 4x + 8.\]
After distributing, we observe that the left-hand side and the right-hand side of the equation are identical: \(4x + 8 = 4x + 8\). This means that for any real number \(x\), the value of both sides will always be equal. Subtracting \(4x\) from both sides gives \(8 = 8\), which is a true statement regardless of the value of \(x\). Since the equation reduces to a statement that is always true, it is an identity. Consequently, the equation is satisfied by all real numbers.
This type of equation is an example of an identity, which is a mathematical statement that is true for every possible value of the variable. In real-life scenarios, identities are frequently used in engineering and physics to simplify complex systems; for instance, when an expression for the total energy of a closed system reduces to a constant regardless of the configuration of its components, it confirms that the energy is conserved. Understanding this concept helps in identifying when two seemingly different expressions represent the exact same state or quantity.