Determine the roots of each equation algebraically. Make sure you do case 1 and case 2
4x +1 = |2x - 3|
To solve the equation 4x + 1 = |2x - 3|, we need to consider two cases: when the expression inside the absolute value is nonnegative and when it is negative.
Case 1: (2x - 3) ≥ 0
When (2x - 3) ≥ 0, the expression inside the absolute value becomes (2x - 3), and the equation simplifies to 4x + 1 = 2x - 3.
Subtracting 2x from both sides, we get:
2x + 1 = -3
Subtracting 1 from both sides, we get:
2x = -4
Dividing by 2, we find:
x = -2
So, x = -2 is a root when (2x - 3) ≥ 0.
Case 2: (2x - 3) < 0
When (2x - 3) < 0, the expression inside the absolute value becomes -(2x - 3), and the equation changes to 4x + 1 = -(2x - 3).
Expanding the negative sign, we have:
4x + 1 = -2x + 3
Adding 2x to both sides, we get:
6x + 1 = 3
Subtracting 1 from both sides, we find:
6x = 2
Dividing by 6, we find:
x = 1/3
So, x = 1/3 is a root when (2x - 3) < 0.
In summary, the roots of the equation 4x + 1 = |2x - 3| are x = -2 and x = 1/3.