Solve $-\frac{1}{2}(4-2 x)=x+8$ for $x$.
Answer (2)
Distribute − 2 1 on the left side: − 2 + x = x + 8 .
Subtract x from both sides: − 2 = 8 .
Since − 2 = 8 is false, there is no solution.
The equation has no solution .
Explanation
Problem Analysis We are given the equation − 2 1 ( 4 − 2 x ) = x + 8 and asked to solve for x . We will simplify the equation by distributing the − 2 1 on the left side, combining like terms, and isolating x to find its value.
Distribute First, distribute the − 2 1 on the left side of the equation: − 2 1 ( 4 − 2 x ) = − 2 1 ( 4 ) − 2 1 ( − 2 x ) = − 2 + x So the equation becomes: − 2 + x = x + 8
Isolate x Next, we want to isolate x . Subtract x from both sides of the equation: − 2 + x − x = x + 8 − x This simplifies to: − 2 = 8
Solve for x Since − 2 = 8 is a false statement, there is no value of x that will make the original equation true. Therefore, the equation has no solution.
Final Answer The equation simplifies to − 2 = 8 , which is a contradiction. Therefore, there is no solution for x .
Examples
Imagine you're trying to balance a seesaw. On one side, you have a weight that changes with a variable x , and on the other side, you have a fixed weight. Solving an equation like this helps you find the exact value of x needed to balance the seesaw. If the equation has no solution, it means no matter what you do with x , you can't balance the seesaw!
After simplifying the equation, we find that it leads to a contradiction: − 2 = 8 . This indicates that there is no solution for x . The equation ultimately has no values that can make it true.
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