While the concept of an interval may be relatively straightforward, every number between one number and another one, it has several uses in mathematics. Intervals can serve labeling purposes on graphs, or as the solution to both equations and inequalities. When an equation has a variable, there could be multiple correct answers. Intervals can describe the set of numbers for which an equation will work, such as every number between zero and three. Intervals can also describe even larger answer sets, such as an equation being true for every number except those that fall between zero and three.
In graphs that describe the frequency of a particular number in a data set, such as a histogram, intervals describe the set of numbers that a particular frequency number is describing. For example, in a data set that gathered the age of everyone who visited a particular movie theater on a particular night, each interval on the graph will describe a particular set of ages. The first interval on the movie-goer histogram might be 0-10, so the number this interval reaches on the y-axis describes the total number of infants through 10-year-olds that were in the theater that night. Intervals on these graphs can either be uniform or varied to display a particular set of numbers together for a particular reason, such as age demographics.
For intervals that represent solution sets for a particular equation, a graph interval might take the form of a line graph with particular values shaded over to represent that solution set graphically. For example, the interval negative three to positive three would have two circles on the hash mark for three on both sides of zero, and a dark line connecting them. When both threes are part of the solution set, these circles will be shaded in. If the interval is every number greater than negative three and less than positive three, then these circles would be hollow.
Math students can solve certain kinds of equations, such as polynomial inequalities, by creating graph intervals that will show the correct answer set. The student creates these intervals by graphing the equation, establishing its x- and y-axis intercept points. Students can ascertain the interval for which a given polynomial equation is positive or negative by the x-values for which the graph has a positive or negative y-value. The graph will create these intervals for the student to see.