What is the solution for $x$ in the equation?
$\frac{1}{2}-x+\frac{3}{2}=x-4$
A. $x=\frac{1}{3}$
B. $x=3$
C. $x=-3$
D. $x=-\frac{1}{3}$
Answer (2)
Combine constants on the left side: 2 1 + 2 3 = 2 , resulting in 2 − x = x − 4 .
Add x to both sides: 2 = 2 x − 4 .
Add 4 to both sides: 6 = 2 x .
Divide both sides by 2: x = 3 . The solution is 3 .
Explanation
Problem Analysis We are given the equation 2 1 − x + 2 3 = x − 4 and asked to solve for x . Our goal is to isolate x on one side of the equation.
Combining Constants First, we combine the constant terms on the left side of the equation: 2 1 + 2 3 = 2 4 = 2 . So the equation becomes 2 − x = x − 4 .
Adding x to Both Sides Next, we want to get all the x terms on one side of the equation. We can add x to both sides: 2 − x + x = x − 4 + x , which simplifies to 2 = 2 x − 4 .
Adding 4 to Both Sides Now, we want to isolate the x term. We can add 4 to both sides of the equation: 2 + 4 = 2 x − 4 + 4 , which simplifies to 6 = 2 x .
Dividing by 2 Finally, we divide both sides by 2 to solve for x : 2 6 = 2 2 x , which simplifies to 3 = x . Therefore, x = 3 .
Final Answer The solution is x = 3 , which corresponds to option B.
Examples
In real life, solving linear equations like this can help you determine how much of an ingredient you need in a recipe, calculate distances, or manage your budget. For example, if you know you have a certain amount of money and want to buy something that costs a certain amount per unit, you can use a linear equation to figure out how many units you can afford.
The solution to the equation 2 1 − x + 2 3 = x − 4 is x = 3 . Thus, the correct answer is option B.
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