Which is the graph of the inequality?

To graph a linear inequality in two variables (say, x and y ), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line.

If the inequality is strict ( < or > ), graph a dashed line. If the inequality is not strict ( ≤ or ≥ ), graph a solid line.

Finally, pick one point that is not on either line ( ( 0 , 0 ) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. 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.

Example 1:

Solve the system of inequalities by graphing:

y ≤ x − 2 y > − 3 x + 5

First, graph the inequality y ≤ x − 2 . The related equation is y = x − 2 .

Since the inequality is ≤ , not a strict one, the border line is solid.

Graph the straight line.

Which is the graph of the inequality?

Consider a point that is not on the line - say, ( 0 , 0 ) - and substitute in the inequality y ≤ x − 2 .

0 ≤ 0 − 2 0 ≤ − 2

This is false. So, the solution does not contain the point ( 0 , 0 ) . Shade the lower half of the line.

Which is the graph of the inequality?

Similarly, draw a dashed line for the related equation of the second inequality y > − 3 x + 5 which has a strict inequality. The point ( 0 , 0 ) does not satisfy the inequality, so shade the half that does not contain the point ( 0 , 0 ) .

Which is the graph of the inequality?

The solution of the system of inequalities is the intersection region of the solutions of the two inequalities.

Which is the graph of the inequality?

Example 2:

Solve the system of inequalities by graphing:

2 x + 3 y ≥ 12 8 x − 4 y > 1 x < 4

Rewrite the first two inequalities with y alone on one side.

3 y ≥ − 2 x + 12 y ≥ − 2 3 x + 4 − 4 y > − 8 x + 1 y < 2 x − 1 4

Now, graph the inequality y ≥ − 2 3 x + 4 . The related equation is y = − 2 3 x + 4 .

Since the inequality is ≥ , not a strict one, the border line is solid.

Graph the straight line.

Consider a point that is not on the line - say, ( 0 , 0 ) - and substitute in the inequality.

0 ≥ − 2 3 ( 0 ) + 4 0 ≥ 4

This is false. So, the solution does not contain the point ( 0 , 0 ) . Shade upper half of the line.

Which is the graph of the inequality?

Similarly, draw a dashed line of related equation of the second inequality y < 2 x − 1 4 which has a strict inequality. The point ( 0 , 0 ) does not satisfy the inequality, so shade the half that does not contain the point ( 0 , 0 ) .

Which is the graph of the inequality?

Draw a dashed vertical line x = 4 which is the related equation of the third inequality.

Here point ( 0 , 0 ) satisfies the inequality, so shade the half that contains the point.

Which is the graph of the inequality?

The solution of the system of inequalities is the intersection region of the solutions of the three inequalities.

Which is the graph of the inequality?

Graphing Inequalities

To graph an inequality, treat the <, , >, or sign as an = sign, and graph the equation. If the inequality is < or >, graph the equation as a dotted line. If the inequality is or , graph the equation as a solid line. This line divides the xy- plane into two regions: a region that satisfies the inequality, and a region that does not.

Next, pick a point not on the line. Check to see if this point satisfies the inequality. If it satisfies the inequality, shade the region which contains that point. If it does not satisfy the inequality, shade the region which does not contain that point. All the points in the shaded region will satisfy the inequality.


Note: The origin (0, 0) is usually the easiest point to test, provided it is not on the line.

Example 1: Graph y≤2x - 3.

Which is the graph of the inequality?
Step 1 -- Graph of y = 2x - 3

Does (0, 0) satisfy y≤2x - 3 ? 0≤2(0) - 3 ? No. Shade the region that does not contain (0, 0):

Which is the graph of the inequality?
y≤2x - 3


Example 2: Graph 3x + 4y < 12.

Which is the graph of the inequality?

Step 1 -- Graph of 3x + 4y = 12

Does (0, 0) satisfy 3x + 4y < 12 ? 3(0) + 4(0) < 12 ? Yes. Shade the region that contains (0, 0):

Which is the graph of the inequality?
3x + 4y < 12


Example 3: Graph y > x.

Which is the graph of the inequality?
Step 1--Graph of y = x

(0, 0) is on the line, so we must pick another point. Does (0, 1) satisfy y > x ? 1 > 0 ? Yes. Shade the region that contains (0, 1):

Which is the graph of the inequality?
y > x

How do you graph in inequalities?

To graph an inequality, treat the <, ≤, >, or ≥ sign as an = sign, and graph the equation. If the inequality is < or >, graph the equation as a dotted line. If the inequality is ≤ or ≥, graph the equation as a solid line.

What does ≥ mean on a graph?

A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤) . The end point is part of the solution. An open circle is used for greater than (>) or less than (<). The end point is not part of the solution.

What is the graph of linear inequality called?

Each line plotted on a coordinate graph divides the graph (or plane) into two half‐planes. This line is called the boundary line (or bounding line). The graph of a linear inequality is always a half‐plane.