How many solutions does the equation |2x − 2| = x have?
Would it be 1
Usually an inequality equation has two solutions ...
|2x-2| = x
2x-2 = x or -2x + 2 = x , where x ≥ 0
x = 2 or -3x = -2
x = 2 or x = 2/3
both answers check out, so there are two solutions.
(2x-2) = x
x = 2
-(2x-2) = x
-2x + 2 = x
3 x = 2
x = 2/3
To find the number of solutions to the equation |2x - 2| = x, we need to consider two different cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: 2x - 2 is positive
In this case, the equation can be rewritten as (2x - 2) = x. Solving this equation gives us x = 2.
Case 2: 2x - 2 is negative
In this case, the equation can be rewritten as -(2x - 2) = x. Simplifying this equation gives us -2x + 2 = x. Solving for x, we get x = 1.
Now, we need to check if the solutions we found satisfy the original equation.
For x = 2:
|2(2) - 2| = 2
|4 - 2| = 2
|2| = 2 (which is true)
For x = 1:
|2(1) - 2| = 1
|2 - 2| = 1
|0| = 1 (which is false)
Therefore, the solution x = 1 does not satisfy the original equation.
So, there is only one solution to the equation |2x - 2| = x, and that is x = 2.