Final answer:
The equation 4(2x+3)=16x+3-8x+9 simplifies to 12=12, which is an identity. Thus, the variable x has an infinite number of solutions, and the solution set is all real numbers.
Step-by-step explanation:
To solve the equation 4(2x+3)=16x+3-8x+9, we first distribute the 4 on the left side of the equation.
8x + 12 = 16x + 3 - 8x + 9
Now, we combine like terms on each side:
8x + 12 = 8x + 12
We can see that the equation is an identity, meaning that any value of x will satisfy the equation. However, if we proceed with the normal steps to solve for x, we'd subtract 8x from both sides and end up with:
12 = 12
This confirms that the equation is true for any x and does not yield a specific solution for x. In cases where you arrive at a statement such as this, where the variable cancels out and you are left with a true statement, it indicates that the solution set is all real numbers.