Questions: Points: Solve and check. Label the equation if it is an identity or a contradiction. a+(a-2)=(a+3)-(a+1) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is a= . (Type an integer or a simplified fraction.) B. The solution is all real numbers. The equation is an identity. C. There is no solution. The equation is a contradiction.

Points: Solve and check. Label the equation if it is an identity or a contradiction. a+(a-2)=(a+3)-(a+1) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is a= . (Type an integer or a simplified fraction.) B. The solution is all real numbers. The equation is an identity. C. There is no solution. The equation is a contradiction.
Transcript text: Points: Solve and check. Label the equation if it is an identity or a contradiction. \[ a+(a-2)=(a+3)-(a+1) \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is $a=$ ]. $\square$ (Type an integer or a simplified fraction.) B. The solution is all real numbers. The equation is an identity. C. There is no solution. The equation is a contradiction.
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Solution

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Solution Steps

To solve the equation \( a + (a - 2) = (a + 3) - (a + 1) \), we will first simplify both sides of the equation. Then, we will solve for \( a \) and determine if the equation is an identity, a contradiction, or has a specific solution.

Step 1: Simplify Both Sides of the Equation

Given the equation: \[ a + (a - 2) = (a + 3) - (a + 1) \]

First, simplify both sides: \[ a + a - 2 = 2a - 2 \] \[ (a + 3) - (a + 1) = a + 3 - a - 1 = 2 \]

So, the equation simplifies to: \[ 2a - 2 = 2 \]

Step 2: Solve for \( a \)

To solve for \( a \), add 2 to both sides of the equation: \[ 2a - 2 + 2 = 2 + 2 \] \[ 2a = 4 \]

Next, divide both sides by 2: \[ a = 2 \]

Step 3: Verify the Solution

Substitute \( a = 2 \) back into the original equation to verify: \[ 2 + (2 - 2) = (2 + 3) - (2 + 1) \] \[ 2 + 0 = 5 - 3 \] \[ 2 = 2 \]

The solution satisfies the original equation.

Final Answer

The correct choice is: \[ \boxed{a = 2} \] The answer is A.

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