The only inequalities where I have to check the values between the points are polynomial and rational inequalities, right?
And not any other type of inequalites
By Checking I mean to know which way to shade on a graph
Just making sure
I reposted because I asked this about 2 hours ago
just in case
Actually, checking the values between the points is common for both polynomial and rational inequalities, but it is not exclusive to these types of inequalities.
In general, when solving any inequality, you need to identify the critical points (where the expression is equal to zero or undefined) and partition the number line into intervals. Then, you choose a test point from each interval to determine the inequality's truthfulness on that interval. Based on the results, you determine which intervals satisfy the inequality.
It is true that polynomial inequalities and rational inequalities often involve checking the intervals between critical points, since these types of inequalities may have multiple solutions and can change signs between specific values. However, there are other types of inequalities, such as radical inequalities or absolute value inequalities, that may also require checking the values between the critical points.
To determine which way to shade on a graph, you can use the test points you identified for the intervals and substitute them into the original inequality. If the test point satisfies the inequality, you shade that region. If the test point does not satisfy the inequality, you shade the opposite region.
Remember, it's important to carefully analyze the inequality and consider all possible cases when solving and graphing any type of inequality.