Solve the equation and check for extraneous solutions.
2|3x+5|=10x+6
To solve the equation, we need to isolate the absolute value on one side. We can divide the equation by 2:
|3x + 5| = 5x + 3
Now, we need to consider two cases:
Case 1: 3x + 5 is positive:
In this case, the equation simplifies to:
3x + 5 = 5x + 3
Subtracting 3x and 3 from both sides, we get:
2 = 2x
Dividing by 2, we find:
x = 1
Case 2: 3x + 5 is negative:
In this case, the equation simplifies to:
-(3x + 5) = 5x + 3
Multiplying both sides by -1, we have:
-3x - 5 = 5x + 3
Adding 3x and 5 to both sides, we obtain:
-8 = 8x
Dividing by 8, we have:
x = -1
Now, we check for extraneous solutions by plugging in x = 1 and x = -1 into the original equation:
For x = 1:
2|3(1)+5|=10(1)+6
2|8| = 16
Since |8| = 8, the equation becomes:
2(8) = 16
16 = 16
The equation is true for x = 1.
For x = -1:
2|3(-1)+5|=10(-1)+6
2|-2| = -10 + 6
Since |-2| = 2, the equation becomes:
2(2) = -4
4 ≠ -4
The equation is not true for x = -1.
Therefore, the only solution to the equation is x = 1.