Solve for $x$:
$2(-x-3)+4x-4=4$
The given equation is \(2(-x-3) + 4x - 4 = 4\). This is a linear equation in one variable, \(x\). The goal is to isolate \(x\) using algebraic manipulation, starting with the distribution property to remove the parentheses, followed by combining like terms and solving for \(x\).
First, we apply the distributive property to the term \(2(-x-3)\):
\[2 \times (-x) + 2 \times (-3) = -2x - 6.\]
Substituting this back into the original equation, we get:
\[-2x - 6 + 4x - 4 = 4.\]
Next, we combine like terms on the left-hand side of the equation.
Group the \(x\) terms:
\[-2x + 4x = 2x.\]
Group the constant terms:
\[-6 - 4 = -10.\]
Substituting these results back into the equation yields:
\[2x - 10 = 4.\]
To isolate the term with \(x\), add 10 to both sides of the equation:
\[2x - 10 + 10 = 4 + 10\]
\[2x = 14.\]
Finally, divide both sides by 2 to solve for \(x\):
\[x = \frac{14}{2} = 7.\]
We have found that \(x = 7\). To verify, we substitute \(x = 7\) back into the original equation:
\[2(-7 - 3) + 4(7) - 4 = 2(-10) + 28 - 4 = -20 + 28 - 4 = 8 - 4 = 4.\]
Since the left side equals the right side, the solution is correct. The final answer is \(\boxed{7}\).
Linear equations are fundamental for modeling real-world situations where rates or costs are constant. For example, if you are planning to rent a specialized tool for a project, the rental company might charge a base fee plus a daily rate. If the total cost for the rental is fixed by your budget, you can use a linear equation to determine exactly how many days you can keep the equipment while staying within your budget. By isolating the variable that represents the number of days, you ensure you manage your resources efficiently.