Solve the equation and check for extraneous solutions.

Question

2|3x+5|=10x+6

Answer 1

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.