GRAPH OF LINEAR FUNCTION

Graphical Solution Of Linear Inequalities In Two Variables

Graphical Solution Of Linear Inequalities In Two Variables

• Show linear inequalities as regions on the coordinate planes.
• To find out if a given point is a solution of a linear inequality.

We make use of inequalities when there is a range of probable answers for a situation like “I have to be there in less than 5 minutes,” “This team is required to score at least a goal to have a chance of winning,” and “To reach the city and come back home again, I require at least N6.50 for train fare” are all instances of situations where a limit is specified, but a range of possibilities exist beyond that limit. That’s what we are wish to study about when we study about inequalities—possibilities.

We can discover the possibilities of an inequality with the use of a number line. This is enough in simple cases, like inequalities with only one variable. But in more composite cases such as those with two variables, it’s more essential to add another dimension, and make use of a coordinate plane. In these situations, we make use of linear inequalities—inequalities that can be written in the form of a linear equation.

One Variable Inequality

Inequalities with one variable can be plotted on a number line, as in the case of the inequality- x ≥ -2 as shown below:

Below is another representation of the same inequality x ≥ -2, this time plotted on a coordinate plane:

On this graph, we first plot the line x = -2, and then shade the whole region to the right of the line. The shaded area is known as the bounded region, and any point within this region satisfies the inequality x≥ -2. Observe as well that the line that stands for the region’s boundary is a solid line; this means that values along the line x = -2 are included in the solution set for this inequality.

As a comparison, look at the graph below, which illustrates y < 3:

In this inequality, the boundary line is plotted as a dashed line. This means that the values on the line y = 3 are not included in the solution set of the inequality.

Observe that the two examples above make use of the variables x and y. It is standard practice to make use of these variables when you are graphing an inequality on a (x, y) coordinate grid.

Two Variable Inequalities

There’s nothing too forceful about the plots of x ≥ -2 and y < 3, as illustrated above. We could have represented both of these relationships on a number line, and depending on the question we were trying to solve, it may have been easier to do so.

Things turn a little more interesting, though, when we plot linear inequalities with two variables. Let’s begin with a basic two-variable inequality: x > y.

The boundary line is shown by a dotted line along x = y. All of the points under the line are shaded; this is the range of points where the inequality x > y is true. Look at the three points that have been found out on the graph. Do you see that the points in the boundary region have x values greater than the y values, whereas the point outside this region do not?

Practice question

When plotted on a coordinate plane, what does the graph of y ≥ x look like?

Plotting other inequalities in standard y = mx + b form is also straight forward. Immediately we graph the boundary line, we can find out which region to shade by testing a few ordered pairs within each region or, in a lot of situations, just by looking at the inequality.

The graph of the inequality y > 4x − 5.5 is illustrated in the figure below. The boundary line is the line y = 4x − 5.5, and it is dashed due to the fact that our y term is “greater than,” not “greater than or equal to.”

To find out the bounded region, the region where the inequality is true, we can test a couple of coordinate pairs, one on each side of the boundary line.

If we substitute (-1, 3) into y > 4x − 5.5, we find 3 > 4(-1) − 5.5, or 3 > -9.5. This is a true statement. It appears like we are required to shade the area to the left side of the line.

On the other hand, if we plug (2, -2) into y > 4x − 5.5, we find -2 > 4(2) − 5.5, or -2 > 2.5. This is not a true statement, therefore, the point (2, -2) ought not to be included in the solution set. Yes, the bounded region is to the left of the boundary line.

Inequalities in Context

Making sense of the significance of the shaded region in an inequality can be a bit taxing without designating any context to it. The following question shows one example where the shaded region assists us to comprehend a range of possibilities.

Cecelia and Jacintha want to donate some money to a local food pantry. To raise funds, they are selling necklaces and earrings that they have made themselves. Necklaces cost N8 and earrings cost N5. What is the range of possible sales they could make in order to donate at least N100?

The first step to follow is to create the inequality. Once we have it, we can solve it and then create a graph of it to better understand the significance of the bounded region. Let’s start by assigning the variable x to the number of necklaces sold and y to the number of earrings sold. (Remember—since this will be mapped on a coordinate plane, we ought to make use of the variables x and y.)

amount of money earned from selling necklaces	+	amount of money earned from selling earrings	≥	$100
8x	+	5y	≥	100

We can rearrange this inequality in order to solve for y. That’s the slope-intercept form, and it will make the boundary line easier to graph.

Example
8x	+	5y	≥	100
8x − 8x	+	5y	≥	100 – 8x
		5y	≥	100 – 8x
		5y⁄5	≥	100 – 8x⁄5
		y	≥	20-8x⁄5

Answer: y = –^8x⁄_{5 + 20}

Therefore, the slope intercept form of the inequality is y = –^8x⁄_{5 + 20} . Now let’s plot the graph of it:

The shaded region stands for all the possible combinations of necklaces and earrings that Cecelia and Jacintha could sell to be able to make at least 100 for the food pantry. It’s really a very broad range.

We can look at the two ordered pairs to confirm that we have shaded the right region. If we substitute (10, 15) into the inequality, we find 8(10) + 5(15) ≥ 100, which is a true statement. Nonetheless, with the use of (5, 5) produces a false statement: 8(5) + 5(5) is just 65, and is therefore less than 100.

Note that while all points will satisfy the inequality, not all points will make sense in this regard. Take for example, (21.25, 10.5), for instance. While it does fall within the shaded region, it’s difficult to expect them to sell 21.25 necklaces and 10.5 earrings! The women can look for whole number combinations in the bounded region to plot how much jewelry to create.

Summary

Inequalities can be mapped on a number line or graphed a coordinate plane. When graphed on a coordinate plane, the full range of possible solutions is shown as a shaded area on the plane. The boundary line for the inequality is drawn as a solid line if the points on the line itself do satisfy the inequality, as in the cases of ≤ and ≥. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of . The use of a coordinate plane is particularly essential for comprehending the range of possible solutions for inequalities with two variables.

The solution of a linear inequality in two variables like Ax + By > C is an ordered pair (x, y) that forms a true statement when the values of x and y are substituted into the inequality.

For instance:
Is (1, 2) a solution to the inequality

The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. The boundary line is dashed for > and < and solid for ≤ and ≥. The half-plane that is a solution to the inequality is normally shaded.

For instance:
Graph the inequality

For two-variable of linear inequalities, the “equals” part is the graph of the straight line; in this case, that means the “equals” part is the line y = 2x + 3:

Now we’re at the point where your textbook possibly gets complex with talk of “test points” and such. But when you did those one-variable inequalities (like x < 3), you didn't take into consideration the "test points"; you merely shaded one side or the other. We can do the same here. Ignore the "test point" matter, and look at the original inequality: y < 2x + 3.

We’ve already graphed the “or equal to” part (it’s merely the line); now we are ready to do the “y less than” part. In other words, this is where we are required to shade one side of the line or the other. Now, let’s consider about it: If I require y LESS THAN the line, do I want ABOVE the line, or BELOW? Of course, I want below the line. Therefore, I shade it in:

And that’s all there is to it: the side I shaded is the “solution region” that is required.

This method is workable due to the fact that we had y alone on one side of the inequality. Almost like with plain old lines, you at all times want to “solve” the inequality for y on one side.

• Graph the solution to 2x– 3y < 6.

First, you’ll solve for y:
2x – 3y < 6
–3y < –2x + 6
y > ( 2/3 )x – 2

[Observe the turned over inequality sign in the last line. You mustn’t forget to turn the inequality over if you multiply or divide through by a negative!]

At this point, you are required to find the “equals” part, which is the line y = ( 2/3 )x – 2.