How many solutions can inequalities have

Let's look at a few examples. Show Solution Check. Related Lessons. Compound Inequalities, Intersections - Definition. View All Related Lessons. Tommy San Miguel. Inequalities with no solution or all real numbers Sometimes, it's possible to solve an inequality and realize it is meaningless! Alex Federspiel. This video by Monica Wieneke shows you some inequalities with no solution or infinite solutions.

Summary You can get inequalities that are false for any variable. Try it! You've reached the end. How can we improve? To find the number of solution sets, the student graphs the inequality, and shades in the values which satisfy each separate inequality. By visually representing the potential values of each one, the student will quickly notice if there is an overlap.

Wherever the shading overlaps is said to be the solution set for the system. If they do not overlap, there is no solution to the system. For example, consider two parallel lines. If the solution to one are the values above the line, and the solution to the other one are the values below the other line, there is no intersection and therefore there is also no solution to the system.

There is another way to use a graphing utility to solve this inequality. In order to satisfy both inequalities, a number must be in both solution sets. So the numbers that satisfy both inequalities are the values in the intersection of the two solution sets, which is the set -2, 4 in interval notation.

Note: When we solved the two inequalities separately, the steps in the two problems were the same. Therefore, the double inequality notation may be used to solve the inequalities simultaneously. In Example 4 above we were looking for numbers that satisfied both inequalities.

Here we want to find the numbers that satisfy either of the inequalities. This corresponds to a union of solution sets instead of an intersection. To make sense of these statements, think about a number line.

The absolute value of a number is the distance the number is from 0 on the number line. This is the set of numbers between -a and a. This means numbers that are either larger than a, or less than -a. Note: -2 and 3 are not in the solution set of the inequality.

We are looking for values of x where the polynomial is negative. The solution set of the inequality corresponds to the region where the graph of the polynomial is below the x-axis.

The critical numbers -2 and 3 are the places where the graph intersects the x-axis. The critical numbers divide the x-axis into three intervals called test intervals for the inequality. We are going to use the fact that polynomial functions are continuous. This means that their graphs do not have any breaks or jumps. Since we have found all the x-intercepts of the graph of x 2 - x - 6, throughout each test interval the graph must be either above the x-axis or below it.