| x- 1 |+| x+3 | = | x- 1 |+| x-(-3)|

(Note: the geometric meaning of |x-a| is the distance from the unknown point "x" to the point "a" on the number axis)

Therefore: | x- 1 |+x-(-3) | represents the sum of the distances from point "x" to "1" and "-3" on the number axis.

Note: On the number axis, the value of x can be to the left of "-3" (that is, x; 1), or between "-3" and "1" (i.e. -3

As can be seen from the number axis:

When x: [1-(-3)]=4;

When -3

Therefore, the minimum value of |x- 1|+|x+3| is 4, and the corresponding value range of x is: -3.

Method 2: (This method is a mathematical method, also known as the zero absolute value method)

Make |x- 1|=0 or | x-3 | = 0；; solve

X= 1 or x=-3.

Note: -3 and 1 divide the number axis into three parts, which is an important basis for our discussion.

(1) When x 4

(2) when x >; At 1, | x-1|+x+3 | = (x-1)+(x+3) = 2x+2 > four.

(3) When -3

Therefore, the minimum value of |x- 1|+|x+3| is 4 and -3.