Graphing Inequalities - Revision Notes

In these revision notes for Graphing Inequalities, we cover the following key points:

• What is the shape and orientation of the first-order equations with one variable?
• The same for the first-order inequalities with one variable.
• How do we identify the direction of the solution set in linear inequalities with one variable?
• The same for linear inequalities with two variables.
• How to find the standard form of a linear inequality when its graph is given?
• What is the shape of a quadratic equation graph?
• How do we identify the direction of the solution set in quadratic inequalities?

Graphing Inequalities Revision Notes

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:

1. In linear inequalities of the form ax + b > 0, the solution set represents all values to the right of the boundary value x = -b/a, without including it.
2. In linear inequalities of the form ax + b < 0, the solution set represents all values to the left of the boundary value x = -b/a, without including it.
3. In linear inequalities of the form ax + b ≥ 0, the solution set represents all values to the right of the boundary value x = -b/a, including it.
4. In linear inequalities of the form ax + b ≤ 0, the solution set represents all values to the left of the boundary value x = -b/a, including it.

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):

The solution set of the inequality y > ax² + bx + c includes the zone above the graph (parabola) but not the graph line.
1. The solution set of the inequality y < ax² + bx + c includes the zone under the graph (parabola) but not the graph line.
2. The solution set of the inequality y ≥ ax² + bx + c includes the zone above the graph (parabola) as well as the graph line.
3. The solution set of the inequality y ≤ ax² + bx + c includes the zone under the graph (parabola) as well as the graph line.