What is the additive inverse of the expression below, where $a$ and $b$ are real numbers?
$2 a+b$
Answer (2)
The additive inverse of an expression x is − x .
The additive inverse of 2 a + b is − ( 2 a + b ) .
Distribute the negative sign: − ( 2 a + b ) = − 2 a − b .
The additive inverse of 2 a + b is − 2 a − b .
Explanation
Understanding the Problem The problem asks us to find the additive inverse of the expression 2 a + b , where a and b are real numbers. The additive inverse of a quantity is the value that, when added to the original quantity, results in zero.
Setting up the Equation To find the additive inverse of 2 a + b , we need to find an expression that, when added to 2 a + b , equals zero. Let's call the additive inverse x . Then we have: ( 2 a + b ) + x = 0
Isolating the Additive Inverse To solve for x , we subtract ( 2 a + b ) from both sides of the equation: x = − ( 2 a + b )
Distributing the Negative Sign Now, we distribute the negative sign to both terms inside the parentheses: x = − 2 a − b
Final Answer Therefore, the additive inverse of 2 a + b is − 2 a − b .
Examples
In real life, additive inverses are useful in balancing equations and financial transactions. For example, if you have a debt of 2 a + b , then having an income of − 2 a − b would perfectly balance your finances, resulting in a net worth of zero. This concept is fundamental in accounting, physics, and engineering, where balancing quantities is essential.
The additive inverse of the expression 2 a + b is − 2 a − b . This value, when added to the original expression, results in zero. Understanding additive inverses is crucial in mathematics, especially in solving equations.
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