# Why isn't $x_2+x_3+x_4\le 2$ a cutting plane?

In my textbook, to generate cutting planes, they tell you to proceed as follows:

A procedure for generating cutting planes:

1. Select a ($$\le$$) constraint that has only nonnegative coefficients.

2. Find a group of variables such that

a) The constraint is violated when all variables in the group equals 1 and the remaining variables equal to 0.

b) But the constraint is satisfied if any one of the variables in the group is changed from 1 to 0.

1. Suppose there are $$K$$ variables in the group add the following constraint as a new cutting plane: Sum of variables in the group $$\le 𝐾 − 1$$

My Question: Why isn't $$x_2+x_3+x_4\le 2$$ a cutting plane? Since if we go by 2a) , and put $$x_2=1 , x_3=1 , x_4=1$$ and $$x_1=0$$ , then it violates the inequality? Since $$11>7$$

It is a cutting plane, but it is implied by $$x_2+x_4\le 1$$ and $$x_3\le 1$$ so not very useful.