Solve the equation for $x$.
$3(x+2)=2(2-x)$
A. $x=-2$
B. $x=-\frac{2}{5}$
C. $x=\frac{1}{2}$
D. $x=3$
Answer (2)
Expand both sides of the equation: 3 x + 6 = 4 − 2 x .
Combine like terms to get 5 x + 6 = 4 .
Isolate the x term: 5 x = − 2 .
Solve for x : x = − 5 2 .
The final answer is − 5 2 .
Explanation
Problem Setup We are given the equation 3 ( x + 2 ) = 2 ( 2 − x ) and we need to solve for x .
Expanding Both Sides First, we expand both sides of the equation: 3 ( x + 2 ) = 3 x + 6 2 ( 2 − x ) = 4 − 2 x So the equation becomes: 3 x + 6 = 4 − 2 x
Combining Like Terms Next, we want to isolate x on one side of the equation. We can add 2 x to both sides: 3 x + 2 x + 6 = 4 − 2 x + 2 x 5 x + 6 = 4
Isolating the x Term Now, subtract 6 from both sides: 5 x + 6 − 6 = 4 − 6 5 x = − 2
Solving for x Finally, divide both sides by 5 to solve for x :
x = 5 − 2 x = − 5 2
Final Answer Therefore, the solution is x = − 5 2 .
Examples
In real-world scenarios, solving linear equations like this can be applied to various situations. For example, imagine you are comparing two different phone plans. Plan A costs $3 per month plus $2 for each gigabyte of data, while Plan B costs $2 per month plus you get 2 gigabytes of data free, but then pay -$2 for each gigabyte you don't use. To find out for what amount of data usage the two plans cost the same, you would set up an equation similar to the one we just solved. By solving for the number of gigabytes, you can determine when the two plans are equally priced, helping you make an informed decision.
The solution to the equation 3 ( x + 2 ) = 2 ( 2 − x ) is x = − 5 2 , which corresponds to option B. The steps involve expanding both sides, combining like terms, isolating the x term, and solving for x .
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