Inequalities Involving Products
Category: Algebra Stuff »
A Plus and Minus Game
Inequalities are always more difficult to solve than their equation counterparts. Even in the world of linear solving where the processes of solving a linear equation and solving a linear inequality are nearly identical, there is an extra step required for the inequalities (flipping the direction of the sign if you multiply or divide by a negative).Once we start solving high degree polynomial equations, it's a lost cause to apply some of the advanced knowledge and theorems to equivalently solve polynomial inequalities. However, if the polynomial can be factored completely when the other side is zero, we can leverage a tool known as sign analysis to find its solution ranges. This technique also works for any type of product on one side of an inequality with zero on the other.Sign Analysis
Consider the following inequality example.$$(x+2)(x-3)(x+3) \gt 0$$Sign analysis can aide us in identifying the intervals that make this inequality true. It's a tool that will work if- One side of the inequality is a product
- The other side is $O$
All in All
In summary, if you have a product, or a polynomial that you can factor into a product, in an inequality with $0$ on the other side, then apply this sign analysis technique to figure out the intervals that make your statement true (which will depend on the direction of your inequality sign). If your inequality is a "less than or equals" ($\leq$) or a "greater than or equals" ($\geq$), make sure to include the endpoints of the interval since those are the boundary numbers that make the product zero. And if you have an inequality with something other than $0$ on the other side, simply subtract that object to the other side so that you do have $0$!- Popular Content Misfit Math »