Similarly, the line separates the plane into two regions. On one side of the line are points with. On the other side of the line are the points with. We call the line a boundary line. The line with equation is the boundary line that separates the region where from the region where. For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not is included in the solution:.
Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in Figure.
In Figure we found that some of the points were solutions to the inequality and some were not. Which of the points we plotted are solutions to the inequality? The points and are solutions to the inequality. Notice that they are both on the same side of the boundary line. The two points and are on the other side of the boundary line , and they are not solutions to the inequality.
For those two points,. What about the point? Because , the point is a solution to the equation. So the point is on the boundary line. Is it a solution to the inequality? Any point you choose on the left side of the boundary line is a solution to the inequality.
All points on the left are solutions. Similarly, all points on the right side of the boundary line, the side with and , are not solutions to. The graph of the inequality is shown in Figure below. The line divides the plane into two regions. The shaded side shows the solutions to the inequality. The points on the boundary line, those where , are not solutions to the inequality , so the line itself is not part of the solution.
We show that by making the line dashed, not solid. The boundary line shown is. Write the inequality shown by the graph. The line is the boundary line. On one side of the line are the points with and on the other side of the line are the points with. At , which inequality is true:. Since, is true, the side of the line with , is the solution. The shaded region shows the solution of the inequality.
Since the boundary line is graphed with a solid line, the inequality includes the equal sign. The graph shows the inequality. We could use any point as a test point, provided it is not on the line. Why did we choose? You may want to pick a point on the other side of the boundary line and check that. Write the inequality shown by the graph with the boundary line. So the side with is the side where. Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.
The graph shows the solution to the inequality. Write the inequality shown by the shaded region in the graph with the boundary line. Graph the linear inequality.
First we graph the boundary line. The inequality is so we draw a dashed line. Then we test a point. Is a solution of? The point is a solution of , so we shade in that side of the boundary line. What if the boundary line goes through the origin? It is in slope—intercept form, with. The inequality is so we draw a solid line.
Now, we need a test point. We can see that the point is not on the boundary line. The point is not a solution to , so we shade in the opposite side of the boundary line. Some linear inequalities have only one variable. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember? It is a horizontal line. Number Line Recall that a number line is a horizontal line that has points which correspond to numbers.
The points are spaced according to the value of the number they correspond to; in a number line containing only whole numbers or integers, the points are equally spaced.
We can graph real numbers by representing them as points on the number line. For example, we can graph " 2 " on the number line: Graph of the Point 2. We can also graph inequalities on the number line. If we pick any number on the dark line and plug it in for x , the inequality will be true.
Upon completing this section you should be able to: Find several ordered pairs that make a given linear equation true. Locate these points on the Cartesian coordinate system. Draw a straight line through those points that represent the graph of this equation. A graph is a pictorial representation of numbered facts. There are many types of graphs, such as bar graphs, circular graphs, line graphs, and so on.
You can usually find examples of these graphs in the financial section of a newspaper. Graphs are used because a picture usually makes the number facts more easily understood. In this section we will discuss the method of graphing an equation in two variables. In other words, we will sketch a picture of an equation in two variables. All possible answers to this equation, located as points on the plane, will give us the graph or picture of the equation.
A sketch can be described as the "curve of best fit. Remember, there are infinitely many ordered pairs that would satisfy the equation. Solution We wish to find several pairs of numbers that will make this equation true. We will accomplish this by choosing a number for x and then finding a corresponding value for y.
A table of values is used to record the data. In the top line x we will place numbers that we have chosen for x. Then in the bottom line y we will place the corresponding value of y derived from the equation.
Of course, we could also start by choosing values for y and then find the corresponding values for x. These values are arbitrary. We could choose any values at all. Notice that once we have chosen a value for x, the value for y is determined by using the equation.
These values of x give integers for values of y. Thus they are good choices. Suppose we chose. We now locate the ordered pairs -3,9 , -2,7 , -1,5 , 0,3 , 1,1 , 2,-1 , 3,-3 on the coordinate plane and connect them with a line. The line indicates that all points on the line satisfy the equation, as well as the points from the table.
The arrows indicate the line continues indefinitely. The graphs of all first-degree equations in two variables will be straight lines. This fact will be used here even though it will be much later in mathematics before you can prove this statement. Such first-degree equations are called linear equations. Equations in two unknowns that are of higher degree give graphs that are curves of different kinds.
You will study these in future algebra courses. Since the graph of a first-degree equation in two variables is a straight line, it is only necessary to have two points. However, your work will be more consistently accurate if you find at least three points. Mistakes can be located and corrected when the points found do not lie on a line. We thus refer to the third point as a "checkpoint. Don't try to shorten your work by finding only two points.
You will be surprised how often you will find an error by locating all three points. Solution First make a table of values and decide on three numbers to substitute for x. We will try 0, 1,2. Again, you could also have started with arbitrary values of y. The answer is not as easy to locate on the graph as an integer would be.
Sometimes it is possible to look ahead and make better choices for x. We will readjust the table of values and use the points that gave integers. This may not always be feasible, but trying for integral values will give a more accurate sketch. We can do this since the choices for x were arbitrary. How many ordered pairs satisfy this equation?
Upon completing this section you should be able to: Associate the slope of a line with its steepness. Write the equation of a line in slope-intercept form. Graph a straight line using its slope and y-intercept. We now wish to discuss an important concept called the slope of a line. Intuitively we can think of slope as the steepness of the line in relationship to the horizontal.
Following are graphs of several lines. Study them closely and mentally answer the questions that follow. If m as the value of m increases, the steepness of the line decreases and the line rises to the left and falls to the right.
In other words, in an equation of the form y - mx, m controls the steepness of the line. In mathematics we use the word slope in referring to steepness and form the following definition:. Solution We first make a table showing three sets of ordered pairs that satisfy the equation. Remember, we only need two points to determine the line but we use the third point as a check.
Example 2 Sketch the graph and state the slope of. Why use values that are divisible by 3? Compare the coefficients of x in these two equations. Again, compare the coefficients of x in the two equations.
Observe that when two lines have the same slope, they are parallel. The slope from one point on a line to another is determined by the ratio of the change in y to the change in x. That is,. If you want to impress your friends, you can write where the Greek letter delta means "change in. We could also say that the change in x is 4 and the change in y is - 1. This will result in the same line.
The change in x is 1 and the change in y is 3. If an equation is in this form, m is the slope of the line and 0,b is the point at which the graph intercepts crosses the y-axis. The point 0,b is referred to as the y-intercept. If the equation of a straight line is in the slope-intercept form, it is possible to sketch its graph without making a table of values.
Use the y-intercept and the slope to draw the graph, as shown in example 8. First locate the point 0, This is one of the points on the line. The slope indicates that the changes in x is 4, so from the point 0,-2 we move four units in the positive direction parallel to the x-axis. Since the change in y is 3, we then move three units in the positive direction parallel to the y-axis. The resulting point is also on the line. Since two points determine a straight line, we then draw the graph.
Always start from the y-intercept. A common error that many students make is to confuse the y-intercept with the x-intercept the point where the line crosses the x-axis.
To express the slope as a ratio we may write -3 as or. If we write the slope as , then from the point 0,4 we move one unit in the positive direction parallel to the x-axis and then move three units in the negative direction parallel to the y-axis. Then we draw a line through this point and 0,4.
Can we still find the slope and y-intercept? The answer to this question is yes. To do this, however, we must change the form of the given equation by applying the methods used in section Section dealt with solving literal equations. You may want to review that section. Solution First we recognize that the equation is not in the slope-intercept form needed to answer the questions asked. To obtain this form solve the given equation for y.
Sketch the graph of here. Sketch the graph of the line on the grid below. These were inequalities involving only one variable. We found that in all such cases the graph was some portion of the number line.
Since an equation in two variables gives a graph on the plane, it seems reasonable to assume that an inequality in two variables would graph as some portion or region of the plane.
This is in fact the case. To summarize, the following ordered pairs give a true statement. The following ordered pairs give a false statement. If one point of a half-plane is in the solution set of a linear inequality, then all points in that half-plane are in the solution set.
This gives us a convenient method for graphing linear inequalities. To graph a linear inequality 1. Replace the inequality symbol with an equal sign and graph the resulting line.
Check one point that is obviously in a particular half-plane of that line to see if it is in the solution set of the inequality. If the point chosen is in the solution set, then that entire half-plane is the solution set. If the point chosen is not in the solution set, then the other half-plane is the solution set.
Why do we need to check only one point? Step 3: The point 0,0 is not in the solution set, therefore the half-plane containing 0,0 is not the solution set. Since the line itself is not a part of the solution, it is shown as a dashed line and the half-plane is shaded to show the solution set.
The solution set is the half-plane above and to the right of the line. Step 3: Since the point 0,0 is not in the solution set, the half-plane containing 0,0 is not in the set. Hence, the solution is the other half-plane. Therefore, draw a solid line to show that it is part of the graph.
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