Solve the equation: |6x - 2| = |3x + 1|
6x-2 = 3x+1 OR 6x-2 = -3x - 1
3x= 3 OR 9x = 1
x = 1 OR x = 1/9
Thanks Reiny
Recall that
|n| = n if n >= 0
|n| = -n if n < 0
So, you have four cases to consider:
(6x-2) >= 0
(6x-2) < 0
(3x+1) >= 0
(3x+1) < 0
In pairing up the conditions, some may be incompatible.
On the other hand, while the expressions may be positive or negative, their square will always be positive. So, we can just as easily say
(6x-2)^2 = (3x+1)^2
9x^2 - 10x + 1 = 0
(x-1)(9x-1) = 0
x = 1/9 or 1
This makes sense, since you rec all the graphs are v-shapes sitting side by side. Since one is steeper than the other, they may well cross in two places.
See that graphs at
http://www.wolframalpha.com/input/?i=solve+|6x+-+2|+%3D+|3x+%2B+1|
To solve this equation, we need to consider two cases: when the expressions inside the absolute value bars are either positive or negative.
Case 1: 6x - 2 ≥ 0 and 3x + 1 ≥ 0
When 6x - 2 ≥ 0, we have 6x ≥ 2, which leads to x ≥ 2/6 or x ≥ 1/3.
When 3x + 1 ≥ 0, we have 3x ≥ -1, which leads to x ≥ -1/3.
So in this case, the solution set is x ≥ 1/3.
Case 2: 6x - 2 < 0 and 3x + 1 < 0
When 6x - 2 < 0, we have 6x < 2, which leads to x < 2/6 or x < 1/3.
When 3x + 1 < 0, we have 3x < -1, which leads to x < -1/3.
So in this case, the solution set is x < -1/3.
Now, let's combine the solutions from both cases.
The solution set for the equation |6x - 2| = |3x + 1| is given by the range where both cases overlap. Hence, the solution set is x < -1/3 and x ≥ 1/3.