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.
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.
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 ) .
The solution of the system of inequalities is the intersection region of the solutions of the two inequalities.
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.
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 ) .
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.
The solution of the system of inequalities is the intersection region of the solutions of the three inequalities.
This is a graph of a linear inequality:
The inequality y ≤ x + 2
You can see the y = x + 2 line, and the shaded area is where y is less than or equal to x + 2
Linear Inequality
A Linear Inequality is like a Linear Equation (such as y = 2x+1) ...
... but it will have an Inequality like <, >, ≤, or ≥ instead of an =.
How to Graph a Linear Inequality
First, graph the "equals" line, then shade in the correct area.
There are three steps:
- Rearrange the equation so "y" is on the left and everything else on the right.
- Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)
- Shade above the line for a "greater than" (y> or y≥)
or below the line for a "less than" (y< or y≤).
Let us try some examples:
Example: y≤2x-1
1. The inequality already has "y" on the left and everything else on the right, so no need to rearrange
2. Plot y=2x-1 (as a solid line because y≤ includes equal to)
3. Shade the area below (because y is less than or equal to)
Example: 2y − x ≤ 6
1. We will need to rearrange this one so "y" is on its own on the left:
Start with: 2y − x ≤ 6
Add x to both sides: 2y ≤ x + 6
Divide all by 2: y ≤ x/2 + 3
2. Now plot y = x/2 + 3 (as a solid line because y≤ includes equal to)
3. Shade the area below (because y is less than or equal to)
Example: y/2 + 2 > x
1. We will need to rearrange this one so "y" is on its own on the left:
Start with: y/2 + 2 > x
Subtract 2 from both sides: y/2 > x − 2
Multiply all by 2: y > 2x − 4
2. Now plot y = 2x − 4 (as a dashed line because y> does not include equals to)
3. Shade the area above (because y is greater than)
The dashed line shows that the inequality does not include the line y=2x-4.
Two Special Cases
You could also have a horizontal or vertical line: