(4x+1)(3-6x)/4x+12<0

To solve the inequality (4x+1)(3-6x) / (4x+12) < 0, we can follow these steps:

Step 1: Factorize the expression (4x+1)(3-6x): (4x+1)(3-6x) = 4x(3) - 6x(3) + 4x(1) - 1(6x) = 12x - 18x + 4x - 6x = -8x

Step 2: Simplify the expression: -8x / (4x + 12) < 0

Step 3: Determine the critical points: Setting the numerator and denominator separately equal to zero, we have: -8x = 0, which implies x = 0 4x + 12 = 0, which implies x = -3

Step 4: Create a number line with the critical points: -∞ --------- -3 --------- 0 --------- +∞

Step 5: Test a value in each interval to determine the sign of the expression:

• Test x = -4: Plug it into the expression: -8(-4) / (-4(-4)+ 12) = 32 / 4 = 8 > 0
• Test x = -2: Plug it into the expression: -8(-2) / (-4(-2)+ 12) = 16 / 8 = 2 > 0
• Test x = 1: Plug it into the expression: -8(1) / (-4(1)+ 12) = -8 / 8 = -1 < 0
• Test x = 2: Plug it into the expression: -8(2) / (-4(2)+ 12) = -16 / 4 = -4 < 0

Step 6: Determine the solution: From the number line, we see that the expression is negative when x < -3 or 0 < x < 1. Thus, the solution to the inequality is -∞ < x < -3 or 0 < x < 1.