Find the smallest value of the function y = | 4x + 6 | +3 | x + 1 |

1. The characteristic points of the function are x = -1.5 and x = -1. These are the points at which the terms | 4 * x +6 | and | x + 1 |.

2. Let’s expand the absolute value modules by scribbling the function by intervals:

y (x) = -7 * x – 9 for x <= – 1.5;
y (x) = x + 3 for -1.5 <= x <= -1;
y (x) = 7 * x + 9 for x> = -1.
3. On the first interval, the minimum value of the function at x = -1.5 is equal to y = 10.5 – 9 = 1.5.

4. On the second interval, the minimum value at x = -1.5 is y = -1.5 + 3 = 1.5.

5. On the third interval, the minimum value at x = -1 is y = -7 + 9 = 2.

Answer: the smallest value of the function is 1.5 at x = -1.5.