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
x+1/2-2x-4/3<1/6
Yes, you are correct. When dealing with polynomial and rational inequalities, you need to check the values between the critical points to determine the direction in which to shade on a graph.
For polynomial inequalities, you typically start by factoring the polynomial and finding the critical points where the expression equals zero. These critical points divide the number line into intervals. You can then choose test points from each interval and plug them back into the inequality. By evaluating the inequality at these test points, you can determine which intervals satisfy the inequality and shade accordingly.
Similarly, for rational inequalities, you begin by finding the critical points where the rational expression is equal to zero or is undefined. These critical points also divide the number line into intervals. After choosing test points from each interval, you evaluate the inequality at these test points to determine the intervals that satisfy the inequality.
While there are other types of inequalities, such as absolute value inequalities or exponential inequalities, where you may need to consider the direction of shading, these inequalities typically do not involve checking values between points like polynomial and rational inequalities do.