Name for Variable satisfying an inequality with equality

In mathematics we often have statements like a x <= b, where a and b are constants and x is a variable.

Now there may be variables satisfying the inequality (that is the statement is true) as well as variables violating it (in which case it's false).

However I don't have any words in my vocabulary to express stricter statements: Is there a word to describe that the variable x satisfies the equality a x = b? What about the case that a x < b? I remember having read "x is a root of the inequality" meaning a x = b somewhere, but I can't remember where.

The equation

ax = b

is a special case of the inequality

ax is less-than-or-equal-to b

(I can't achieve the correct symbol either).

If x satisfies the equation, it is a root of the equation or a solution of the equation. (in fact, for a linear equation as here, the root / solution)

If x satisfies the inequality (here extended to mean the combination of equation / inequality), it is a member of the solution set of the inequality. This will typically be an infinite set. I've never heard of a term like 'root' being used in this case.

In the domain of Operations Research, the variable that you add to make an inequality an equality is a slack variable. I realize that you are looking for a factor to multiply rather than an augend to add, but the "slack variable" synapse fired for me.