# 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.

• Wouldn't this sort of specialised vocabulary be more likely to be found on Math.SE? Nov 12 '13 at 14:03
• @AndrewLeach Well maybe it's known there, but it's specific to the english language. That's why I asked it here. Nov 12 '13 at 14:04
• I've seen those Math folks on Math.SE write above-average English. They can format math better than here, as well. Further, you'd also have the opportunity to find out if there's a German phrase if there were no English equivalent. Nov 12 '13 at 14:30
• Well then, if you all think it's better there, I won't stop you ;-) May this be migrated easily or shall I just post a new question there? Nov 12 '13 at 14:35
• You can flag for mod attention yourself and ask for it to be migrated. Do not post the same question on both sites please. For what it's worth, I think the question is fine here, I'd wait a while to see if you get a good answer and then ask for migration of you don't. Nov 12 '13 at 14:41

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.

• Nope, that's not remotely what I'm looking for. I do not have factors or additional variables. It's the property of the variable I'm interested in. Nov 12 '13 at 13:55

In mathematics, if an inequality is satisfied with equality, you say that the inequality is tight.

• Interesting and certainly a step in the right direction. However I'm searching for the term describing the variable that makes the inequality tight. Nov 12 '13 at 22:10
• I don't know if there's a name for this, because it isn't really a valid mathematical concept. There's not just one variable that makes the inequality tight … if there are four variables involved in the inequality, they all make the inequality tight together. Change any one of them, and the two sides are probably no longer equal. If there's only one variable in the inequality, then it does make sense mathematically, but I don't know a name for it. Nov 12 '13 at 22:43
• I never required uniqueness. In fact, this would be rather odd. I also never stated that this is a one-dimensional problem. I'll rephrase the question later. Nov 13 '13 at 7:58
• I think I now understand; you're looking for a short term for "the values of the variables that make the inequality tight". I don't know whether such a term exists. Nov 13 '13 at 8:34