Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables. Represent the Cartesian coordinate system and identify the origin and axes. Given an ordered pair, locate that point on the Cartesian coordinate system.

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.May 20, · regardbouddhiste.comg an inequality from a graph Mr.

McLaughlin Web. Writing an inequality from a graph Write linear inequalities from a graph - Duration. Since the entire graph gets covered, we say that the solution set is (−∞,∞).

-2 0 2 4 6 (] R.2 Exercises Write the interval notation for the inequality. 1. x ≥4 2.

23 terms. MrJSmith2. Solving and Graphing Inequalities. Make sure that you can draw a graph from an inequality and write an inequality when given a graph. Pay attention to open and closed circles. STUDY. PLAY. x>5. What is the inequality for the graph? x. In the Java Grapher the expression (*x)L3 has the value 1 for numbers x that satisfy the inequality, and the value 0 for other numbers x. The picture below shows the graph of (*x. The graph of a compound inequality with an "and" represents the intersection of the graph of the inequalities. A number is a solution to the compound inequality if the number is a solution to both inequalities. It can either be written as x > -1 and x.

−4 6. −7 ≤x x 9. −10 2 x ≤−1 or x >1 Solve and graph. Inequalities and Their Graphs Date_____ Period____ Draw a graph for each inequality. Write an inequality for each graph. 21) a 22) x 23).

The graph of an inequality shows all of the solutions of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction.

EXAMPLE 3 .

POINTS ON THE PLANE |
Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables. |

What happens if we multiply both numbers by the same value c? |

In an inequality, one side of the inequality can be larger or smaller than the quantity on the other side. The math symbols, and ≥ provide information about the relative sizes of the two expressions.

Practice: Two-variable inequalities from their graphs. Intro to graphing systems of inequalities.

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Graphing systems of inequalities. Practice: Systems of inequalities graphs. Graphing inequalities (x-y plane) review. Next tutorial. Modeling with linear inequalities.

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