- Simplify the right-hand side using the distributive property: \(2(x + 2) = 2x + 4\).
- Compare both sides of the equation: \(2x + 4 = 2x + 4\).
- Recognize that this equation is an identity, true for all values of \(x\).
- Conclude that there are infinitely many solutions: \(\boxed{\text{The equation has infinitely many solutions}}\).
Explanation
- Analyzing and Simplifying the Equation
The equation given is \(2x + 4 = 2(x + 2)\). To analyze the number of solutions, we first simplify the expression on both sides. The left-hand side is \(2x + 4\). The right-hand side is \(2(x + 2)\). By applying the distributive property of multiplication over addition, we multiply 2 by each term inside the parentheses: \(2(x) + 2(2) = 2x + 4\). Now, the equation becomes \(2x + 4 = 2x + 4\).
- Determining the Number of Solutions
By subtracting \(2x\) from both sides of the equation \(2x + 4 = 2x + 4\), we are left with \(4 = 4\). This is a statement that is true for any value of \(x\). Since the equation simplifies to a statement that is always true, it means that any real number substituted for \(x\) will satisfy the original equation. Therefore, the equation has infinitely many solutions.
Examples
In real-world scenarios, identities like this one represent situations where two processes or conditions are equivalent. For example, if two different rental plans charge the same amount regardless of the number of hours used, the cost function for both would look like this. Since the costs are identical, any number of hours you choose will result in the same total cost, showing that there are infinitely many solutions when comparing the two cost models. Understanding when equations are always true helps in simplifying complex business or engineering models into their most basic, equivalent forms.