And expressions cannot be solved. We identify a set of points in the relevant space which are part of the solution set of the equation or inequality. The space may have any number of dimensions, the solution set may be contiguous or in discrete "blobs". There is no solution because there is no equation or inequality ; only an expression.
A solution to a linear inequality in two variables is an ordered pair x, y that makes the inequality a true statement. The solution set is the set of all solutions to the inequality. The solution set to an inequality in two variables is typically a region in the xy-plane, which means that there are infinitely many solutions.
Sometimes a solution set must satisfy two inequalities in a system of linear inequalities in two variables. If it does not satisfy both inequalities then it is not a solution. An inequality determines a region of space in which the solutions for that particular inequality.
For a system of inequalities, these regions may overlap. The solution set is any point in the overlap. If the regions do not overlap then there is no solution to the system. It therefore cannot have a solution so that its solution set is the null set. There is nothing in the solution set - no integers, rationals, irrationals: nothing!
There is no solution set. An empty set is a set with no elements. The solution set for an equation that has no solution is also called an empty set. No, it is part of the solution set. This compound inequality cannot be solved. Log in. Study now. See Answer. Best Answer. 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. This is where we need to know that the graph does not have any breaks. This means that we may choose any number we like in a test interval and evaluate the polynomial at that number to see if the graph is above or below the x-axis throughout that test interval.
When a product of two numbers is equal to 0, then at least one of the numbers must be 0. However, a product of two negative numbers is not negative, so this approach is not useful for solving inequalities. This problem is much more difficult than the inequality in the previous example! It is not easy to factor, so we will not be able to find the exact values of the critical numbers.
We will use a graphing utility to approximate the critical numbers. The graph of the polynomial is shown below.
The critical numbers are approximately One method of solving this problem is to test all the values in the replacement set using a table. Solution sets for inequalities are often infinite sets; we can't list all the numbers. So, we use a special notation.
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