Inequalities Involving Products

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 $O$

Here's how it works.Since we have a product, the left side will only actually be equal to zero when one of its factors is zero. For our example, this happens at $x=-2$, $x=3$, and $x=-3$. Draw a number line and mark these numbers.

Since there is no where else that this product can be zero, the result of this product in each of the four intervals that we divided the number line into will either be positive or negative. And because there are no other zeroes, each interval will always be either positive or negative - we don't have to worry that the result might change from one to another within the same interval.

All we have to do is test out an $x$ value in each interval to know whether a particular interval will give positive or negative results.Interval one - from $-\infty$ to $-3$:Let's pick $x=-4$.$$((-4)+2)((-4)-3)((-4)+3)$$$$=(-)(-)(-) = (-)$$Notice that we don't even care what the actual numeric answer is - we just want to know whether it's positive or negative. Feel free to compute the actual result if it helps!

Now that we know the first interval yields negative results, let's move on to the next.Interval two - from $-3$ to $-2$:Let's pick $x=-2.5$.$$((-2.5)+2)((-2.5)-3)((-2.5)+3)$$$$=(-)(-)(+)=(+)$$

And so on. Testing the other two intervals (interval three from $-2$ to $3$ and interval four from $3$ to $\infty$) will give us the complete picture. Check on your own and see if you can match the final result.

For this example, since we were asked to find when $(x+2)(x-3)(x+3) \gt 0$, or in words, when the expression is positive, we know from our sign diagram that this will be true when $x$ is either between $-3$ and $-2$, or between $3$ and $\infty$.

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 $0$!