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

Question

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.

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

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 $11>7$

Answer 1

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.