What is the value of [tex]$x$[/tex] in the equation [tex]$2(x+3)=4(x-1)$[/tex]?
Answer (2)
The solution to the equation 2 ( x + 3 ) = 4 ( x − 1 ) is x = 5 . We reached this solution by expanding, isolating, and solving for x . Finally, we verified our answer by substituting it back into the original equation.
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Expand both sides of the equation: 2 x + 6 = 4 x − 4 .
Isolate x by subtracting 2 x from both sides and adding 4 to both sides: 10 = 2 x .
Divide both sides by 2 to solve for x : x = 5 .
Verify the solution by substituting x = 5 into the original equation, which holds true, so the final answer is 5 .
Explanation
Understanding the Problem We are given the equation 2 ( x + 3 ) = 4 ( x − 1 ) and we want to find the value of x that satisfies this equation.
Expanding Both Sides First, we expand both sides of the equation by distributing the constants: 2 ( x + 3 ) = 2 x + 6 4 ( x − 1 ) = 4 x − 4 So the equation becomes: 2 x + 6 = 4 x − 4
Isolating x Next, we want to isolate x on one side of the equation. We can subtract 2 x from both sides: 2 x + 6 − 2 x = 4 x − 4 − 2 x 6 = 2 x − 4
Further Isolating x Now, we add 4 to both sides of the equation: 6 + 4 = 2 x − 4 + 4 10 = 2 x
Solving for x Finally, we divide both sides by 2 to solve for x :
2 10 = 2 2 x 5 = x So, x = 5 .
Verifying the Solution To verify our solution, we substitute x = 5 back into the original equation: 2 ( 5 + 3 ) = 4 ( 5 − 1 ) 2 ( 8 ) = 4 ( 4 ) 16 = 16 Since the equation holds true, our solution is correct.
Final Answer Therefore, the value of x in the equation 2 ( x + 3 ) = 4 ( x − 1 ) is 5.
Examples
In real life, this type of equation can be used to solve problems involving comparisons of costs or quantities. For example, suppose you are comparing two different phone plans. Plan A costs $2 per month plus $3 per gigabyte of data, while Plan B costs $4 per month but only $1 per gigabyte. The equation 2 ( x + 3 ) = 4 ( x − 1 ) can represent the number of gigabytes, x , for which the total cost of the two plans is the same. Solving the equation helps you determine when the two plans are equally cost-effective.
