Solve for x:
$3(x-4)+2=x+2(x-5)$
The given equation is \(3(x - 4) + 2 = x + 2(x - 5)\). We want to determine the value of \(x\) that satisfies this equality. To do this, we simplify both sides of the equation by applying the distributive property, which allows us to multiply the terms inside the parentheses by the coefficient outside. After expanding, we will collect like terms to see if a single solution for \(x\) exists or if the equation holds for all values.
First, we expand the left side of the equation:
\[3(x - 4) + 2 = 3 \cdot x - 3 \cdot 4 + 2 = 3x - 12 + 2.\]
Combining the constant terms \(-12\) and \(+2\), we get:
\[3x - 10.\]
Next, we expand the right side of the equation:
\[x + 2(x - 5) = x + 2 \cdot x - 2 \cdot 5 = x + 2x - 10.\]
Combining the like terms \(x\) and \(2x\), we get:
\[3x - 10.\]
Thus, the equation simplifies to:
\[3x - 10 = 3x - 10.\]
We have arrived at the result \(3x - 10 = 3x - 10\). This statement is an identity, meaning it is true for any real number \(x\). If we were to subtract \(3x\) from both sides, we would get \(-10 = -10\), which is always true regardless of the value of \(x\). Therefore, the equation has infinitely many solutions; any real number is a valid solution. We can express this as \(x \in \mathbb{R}\) or all real numbers.
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