Determine which point is a solution to the inequality

If they do, shade the half-plane containing that point. If they don't, shade the other half-plane. Graph each of the inequalities in the system in a similar way. The solution of the system of inequalities is the intersection region of all the solutions in the system.

This is false. So, the solution does not contain the point 0 , 0. Shade the lower half of the line. The point 0 , 0 does not satisfy the inequality, so shade the half that does not contain the point 0 , 0. Makes the inequality.

If substituting x, y into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region , or the point will be part of a dotted boundary line. Use the graph to determine which ordered pairs plotted below are solutions of the inequality.

Solutions will be located in the shaded region. These values are located in the shaded region, so are solutions. These values are not located in the shaded region, so are not solutions. The correct answer is 3, 3. This is a true statement, so it is a solution to the inequality. Graphing Inequalities.

Plotting inequalities is fairly straightforward if you follow a couple steps. Shade the region that contains the ordered pairs that make the inequality a true statement. You can use the x - and y - intercepts for this equation by substituting 0 in for x first and finding the value of y ; then substitute 0 in for y and find x. Plot the points 0, 1 and 4, 0 , and draw a line through these two points for the boundary line. The next step is to find the region that contains the solutions.

Is it above or below the boundary line? To identify the region where the inequality holds true, you can test a couple of ordered pairs, one on each side of the boundary line. This is a false statement, since 11 is not less than or equal to 4. This is true! The region that includes 2, 0 should be shaded, as this is the region of solutions. 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. The solution set is the line and the half-plane below and to the right of the line. Next check a point not on the line.

Notice that the graph of the line contains the point 0,0 , so we cannot use it as a checkpoint. The point - 2,3 is such a point.

When the graph of the line goes through the origin, any other point on the x- or y-axis would also be a good choice. Upon completing this section you should be able to: Sketch the graphs of two linear equations on the same coordinate system. Determine the common solution of the two graphs. Example 1 The pair of equations is called a system of linear equations. We have observed that each of these equations has infinitely many solutions and each will form a straight line when we graph it on the Cartesian coordinate system.

We now wish to find solutions to the system. In other words, we want all points x,y that will be on the graph of both equations.