Final answer:
The correct answer is Option A, where the equation simplifies to an identity (x - 20 = x - 20), indicating an infinite number of solutions, as any value of x satisfies the equation.
Step-by-step explanation:
The question asks which equation has an infinite number of solutions. An equation has an infinite number of solutions when both sides are identical after simplification, meaning any value for the variable will satisfy the equation.
Let's examine each option:
• A. 3x - 2(x + 10) = x - 20: Simplifying this gives 3x - 2x - 20 = x - 20, which simplifies to x - 20 = x - 20. This suggests an infinite number of solutions, but let's check the other options to confirm.
• B. 5x + 2(x - 3) = 5x + 2(3 - x): Simplifying yields 5x + 2x - 6 = 5x + 6 - 2x, which simplifies further to 7x - 6 = 3x + 6. This is not an identity; therefore, it does not have an infinite number of solutions.
• C. x/2 + 1 = 3x/10 + 3: Simplifying gives 5/10x + 1 = 3/10x + 3, which further simplifies to 2/10x = 2, allowing for the determination of a specific solution for x. Thus, it does not have an infinite number of solutions.
• D. 7/2x + x = 7/4: Simplifying this equation gives 7/2x + 2/2x = 7/4, which further simplifies to 9/2x = 7/4. This will yield a specific value for x and therefore does not have an infinite number of solutions.
After reviewing all the options, the correct answer is Option A because the equation reduces to an identity, indicating that any value of x will satisfy it.