Introduction to inequalities

Firstly,lets review what should be familiar — the arithmetic of equations. Suppose A = B and P = Q. The soundbyte is: you can combine them in almost any way imaginable to get a new valid equation. You can add them, in either order (A + P = B + Q) or (A + Q = B + P). You can subtract then, either one from the other, in either order (four subtraction equations: e.g. A – P = B – Q). You can multiply them in either order (A* P = B * Q) or (A * Q = B * P). You can divide either one by either other (assuming you are not dividing by zero), in either order (four division equations: e.g. A/P = B/Q). Things get considerably more interesting if some or all of those letters are not individual numbers but algebraic expressions. Adding inequalities Everything gets trickier with inequalities. First of all, an equation, such as A = B, is inherently symmetrical and “two-sided”, but an inequality is more a one-sided, unidirectional thing. With any arithmetic of inequalities, we must consider the direction of the inequality. Adding inequalities is not so bad: you can add inequalities with the same direction of inequality. Thus, if A > B, and P > Q, then it must be true that A + P > B + Q. That always works, and it is in many ways what you’d expect. Subtracting inequalities This is the one that’s much trickier. If A > B, and P > Q, then naïvely one might expect that (A – P) would be greater than (B – Q), but that’s not necessarily the case. For example, suppose we have 10 > 5 and 2 > 1 — then we could subtract them in the same direction of inequality, and we’d get 8 > 4, which still works. BUT, suppose 10 > 5, and 100 > 1, both true — now, if we subtract in the same direction of inequality, we find that (–90) is not greater than 4.We can’t subtract inequalities with the same direction of inequality, but we CAN subtract inequalities with the opposite direction of inequality — in other words, if A > B, and P > Q, then it must be true that (A – Q) > (B – P). Multiplication and division Everything gets much hairier with multiplying or dividing inequalities when you consider — one or both sides could be negative. If we multiply or divide both sides of inequality by a single negative number, that’s perfectly legal, but we must remember to reverse the direction of inequality. What happens if we were to multiply or divide inequalities and negatives were involved? For example, if we know that x > –6 and y > –7, then what can we say about the product xy? As it turns out, we could pick an x and a y that would satisfy the original inequalities and make the product xy equal absolutely any number on the number line. Let focus, though, on a special case.. Suppose we know that all four numbers or expressions are positive: A > B > 0 and P > Q > 0. Then, as with addition, we can multiply inequalities with the same direction: A*P > B*Q must be true. And, as with subtraction, we can divide inequalities with the opposite direction: A/Q > B/P. Again, remember the caveat: everything must be positive for these patterns to work. Absolute value inequalities Distance in the geometric sense of the word, we don’t care about sign or direction — the distance between two points is just a positive number and is the same, whether from A to B or from B to A. That’s where absolute value comes in. The expression |p – q| is the distance between numbers p & q on the number line.Thus, |x – 5| is the distance between variable point x and fixed point 5. The expression |x – 5| < 2 indicates the set of all points x that have a distance to the point 5 of less than two. Immediately, just thinking about this logic, and without any further calculations, we can see that |x – 5| < 2 is entirely equivalent to 3 < x < 7.     inequality algebra inequality math inequalities graphing inequality examples inequality problems inequality maths number line inequality equations inequality symbols