Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
In addition to the revision notes for Graphing Inequalities on this page, you can also access the following Inequalities learning resources for Graphing Inequalities
Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
---|---|---|---|---|---|---|
10.3 | Graphing Inequalities |
In these revision notes for Graphing Inequalities, we cover the following key points:
The graph of a first-order equation with one variable represents a vertical line, as the graph concerns only the position of the coordinate x in a number line. Hence, when we represent graphically this type of equation on a XY coordinate system, it shows a vertical line because the y-coordinate does not matter. Since the general form of such an equation is ax + b = 0, where the root is calculated by x = -b/a, the solution set of the corresponding inequalities will be one of the four following formulas
The graph line is included only in the last two cases.
From the inequalities above, we obtain the four following rules:
When the variable is not denoted by x but by y insted, the graph will be horizontal. All the above rules are true except the orientation. Thus, if y > -b/a, the solution set includes the part above the graph without the graph line; if y < -b/a, the solution set includes the part below the graph without the graph line; if y ≥ -b/a, the solution set includes the part above the graph including the graph line as well; and if y ≤ -b/a, the solution set includes the part below the graph as well as the graph line.
Linear inequalities in two variables contain two variables at the first power. Their general form is one of the following
where a and b are coefficients, while c is a constant. All of them derive from the linear equation with two variables
The slope of this graph (otherwise known as the "gradient") is obtained by the formula
As we know, another form of writing a linear equation with one variable is to isolate the variable y and write it in terms of the other variable x in the form
where m here represents the gradient k, while n is obtained by the formula
It is better to have the linear inequalities written based on the second form of the corresponding linear equation y = mx + n, as this form allows us to better locate the position of the solution set for that inequality. In this way, we obtain the following four possible linear inequalities with two variables:
Thus, if we have the first linear inequality y > mx + n, the solution set includes all values (the zone) above the graph without the graph line, while in the second inequality y < mx + n the solution set includes all values (the zone) below the graph without the graph line.
On the other hand, the solution set of the third inequality y ≥ mx + n includes all values above the graph as well as those on the graph line, while the solution set of the fourth inequality y ≤ mx + n includes all values below the graph including those of the graph itself.
Sometimes, we have the graph of a linear inequality given but not the inequality shown by that graph. To find the standard form of that inequality, we must first find the corresponding linear equation representing the boundary line of the inequality. For this, we need the coordinates of two known points A and B of the graph to calculate the gradient k by applying the formula
Then, using the equation of the line
we can find the constant n by substituting the coordinates of any from the known points.
Finally, looking at the highlighted region on the graph, you can determine the standard form of the given inequality.
A second-order equation is an extension of the concept of quadratic equations including a new variable y. This means the quadratic equation
is a special case of the second-order equation with two variables
where y = 0. The line that represents the graph of second-order equations with two variables is not straight; this line is called a parabola.
If a second-order equation is expressed in the standard form as the one shown above, the following rules are true for the four corresponding inequalities (the condition is that a > 0):
Enjoy the "Graphing Inequalities" revision notes? People who liked the "Graphing Inequalities" revision notes found the following resources useful:
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Math tutorial "Graphing Inequalities" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.