The Real Number Line. The most intuitive way is to use the real number line. If we draw a line, designate a point on the line to be zero, and choose a scale, then every point on the line corresponds uniquely to a real number, and vice versa: The real number line "respects" the order of the real numbers.
The first step is to find the zeros of the polynomial x2 - x - 6. 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 x2 - 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.
Common Mistake We will use the problem in Example 8 to illustrate a common mistake. 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. In this problem we looking for regions where the graph is above the axis. Return to Contents Rational Inequalities A rational expression is one of the form polynomial divided by polynomial.
In general, graphs of rational functions do have breaks. They are not defined at the zeros of the denominator.
These are the only places where there are breaks, so we can use the same technique to solve rational inequalities that we use for polynomial inequalities. The critical numbers for a rational inequality are all the zeros of the numerator and the denominator.
Since the numerator and denominator are already factored in this example, we see that the critical numbers are -3, 5, and 1. The three critical numbers divide the number line into four test intervals. Since the value is negative, the graph of the rational function is below the x-axis throughout the interval.
The value of the function at 0 is 5, which is positive. The graph of the function is above the x-axis throughout the interval. The value at 2 is The graph of the function is above the x-axis.
We are looking for regions where the graph is above the x-axis, so the solution set is -3, 1 union 5, inf. A graphing utility can be used to see which side of the x-axis the graph is on over the various test intervals.
In some cases you must solve algebraically to find the exact values of the critical numbers, but once this is done, a grapher provides a fast way to finish the problem.
We need to compare an expression to 0. Do not multiply both sides of an inequality by an expression with a variable. For example, given the problemdo not multiply both sides by x. The correct way to handle this problem is as follows: Now we see that the critical numbers are 0 from denominator1, and Definition: A boundary is something that divides one item from another.
Math Usage: The graph of a linear inequality in two variables is a region system of linear inequalities solution of a system of linear inequalities Follow the steps to write a system of inequalities and graph it.
A compound inequality is an equation with two or more inequalities joined together with either "and" or "or" (for example, and; or).When two inequalities are joined with and, they are often written simply as a double inequality, attheheels.com solution of an "and" inequality is the intersection of .
Exercise Set Interval Notation and Linear Inequalities 94 University of Houston Department of Mathematics For each of the following inequalities: (a) Write the inequality algebraically.
A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the graph of all solutions of the system.
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEP T, METHODS, AND STEPS) By Nghi H.
Nguyen DEFINITION. A trig inequality is an inequality in standard form: R(x) > 0 (or. An inequality has no solution will always give you a contradiction, no matter what value you plug in for the variable.
Watch this tutorial, and then try to make your own inequality with no solution! Keywords.