Solve the equation:
$-4(x+2)=-3(x+5)-x$
To solve the equation \(-4(x + 2) = -3(x + 5) - x\), we first expand the expressions on both sides of the equation. Starting with the left side, we distribute the \(-4\) to each term inside the parentheses:
\[-4(x + 2) = -4x - 8\]
Next, we expand the right side, distributing the \(-3\) into the parentheses:
\[-3(x + 5) - x = -3x - 15 - x\]
Now, we combine the like terms on the right side of the equation, which are \(-3x\) and \(-x\):
\[-3x - x = -4x\]
So the expression on the right side simplifies to \(-4x - 15\).
Now that we have expanded both sides, the equation is:
\[-4x - 8 = -4x - 15\]
To solve for \(x\), we try to isolate the terms containing \(x\) on one side. We can add \(4x\) to both sides of the equation:
\[-4x + 4x - 8 = -4x + 4x - 15\]
This simplifies to:
\[-8 = -15\]
Since \(-8\) is never equal to \(-15\), this statement is a contradiction. The original equation has no value of \(x\) that satisfies it. Therefore, there is no solution to the equation.
Equations that result in a contradiction, like the one here, are common in real-world scenarios when trying to find a value that satisfies conflicting constraints. For example, if you are calculating how many hours of work two different contractors would need to bill to reach the exact same total cost, but their base fees and hourly rates lead to parallel cost functions, you would find that their costs never meet. This helps you identify that a solution to balance the costs simply does not exist under the given terms.