Inequalities

Solving Inequalities

One of the most notable examples is that there are multiple types of inequalities: less than, greater than, less than equal to, greater than equal to. The constraints by each of these are self explanatory, but the most important thing to remember is that when solving an equation, you can switch the sign and always need to be conscious of where it is facing. For example, the equation \(4 < x\) is equal to \(x > 4\), but the latter is preffered for standard formatting. In typical equations, it is standard to switch the order of the expressions to conform with standard mathematical best practices, but in inequalities, you need to make sure you are preserving the direction of the inequality with respect to the expression. The process for solving inequalities, in the sense of isolating a variable, is almost exaclty the same as solving an equation. The most notable difference is that when dividing by a negative value, you must switch the direction of the inequality. The reason for this is pretty interesting: the absolute value of a number increases as it gets further from the number line (or is it farther? does math count as physical distance?) but for negative numbers, the actual value decreases. When we multiply by a negative number, we are switching the order of which numbers are bigger, because now the closer you are to the numberline the bigger. This thus forces us to change the sign of a inequality

Squaring Inequalities

Obviously, whenever we square any real number, the result will be positive. However, with inequalities which model the relationships of expressions, this can cause errors. Say you have the equation \(a < b\) for \(a < 0 < b\). Furthermore, let's assume that \(|a| > |b|\). Squaring \(a\) will give a positive value greater than the square of \(b\), but \(a^2 < b^2\) can't be true then. We can't make it a rule to switch the sign, because we cannot always assure that \(|a| > |b|\). Thus, it is generally considered wrong to square an inequality when \(a\) and \(b\) are of different signs.

Another example of how squaring can become troubling in inequalities is with the equation \(x^2 < 4\). If we take the root of both sides, we get \(x < 2\), which albeit one bound for the solutions of the equation, fails to encompass \(x > -2\), demonstrating how when using negative numbers, we also must switch the direction of the inequality. Because of the issues that arise from squaring inequalities, it is important to employ logic when solving inequalities that deal with squares or powers.

Reciprocals

In a way, reciprocals of inequalities are going to be similar to squares, because after all, the reciprocal is just the expression raised to \(-1\). If the signs of the expressions or values on either side of the inequality are the same, you will always have to flip the inequality. The logic for this is similar to that of negatives. For \(\frac{1}{n}\), as \(n\) increases, \(\frac{1}{n}\) decreases.

Modeling

When modeling word problems or other equatoins, it is important to distinguish when to use an equation and when to use an inequality. I will be honest, it is rare to need an inequality to solve something, simply because it is pretty easy to just mentally bound the situations. Inequalities really only come in handy in equations where there are a lot of sign changes. That being said, a good rule of thumb is to use inequalities when modeling anything that does not explicitly say equals. The lack of strictness will also help you model the problem more accurately than how rigid equations are

AM-GM Inequality

The AM-GM Inequality (Arithmetic Mean - Geometric Mean Inequality) states that the arithmetic mean of two numbers will always be greater than the geometric mean. It can be expressed algebraicly as \(\frac{a+b}{2} > \sqrt{ab}\). This often comes in handy on the AMC 10 when you need to find the minimum value of a sum or product of two numbers

Triangle Inequality

Another emphasis that inequalities are closely tied to geometry, it is a well-known inequality that the sum of two side lengths of a triangle must be greater than the remaining side length. This, along with other rules about relationships in triangles, allows you to solve a lot of the similar triangle and complex geometry problems you will find on the AMC 10. It allows you to bind the possible values of a triangle, both in side lengths and area, perimeter, or anything that can be derived from the side lengths

Cauchy Schwarz Inequality

I doubt this will come up on the AMC 10, but if you're interested in theory and want to advance your knowledge, I would look into the Cauchy Schwarz Inequality, which dictates for vectors \(u,v\), the dot product of \(u, v\) will always be \(\leq\) the product of the magnitudes (lengths) of \(u,v\). This very well may never come up on the AMC 10, but it is an important tool in hgiher mathematics for binding the relationship between two vectors and ensuring that their length does not grow past a certain impossible point.