What is the graph of linear inequality?

Q: What is the graph of linear inequality called?

This line is called the boundary line (or bounding line). The graph of a linear inequality is always a half‐plane.

Q: What is the graph of a linear equation?

A linear equation is an equation with two variables whose graph is a line.

Q: What is a linear inequality example?

An example of a linear inequality is 3x-2y> 1. Here, the LHS of the inequality equation is strictly greater than RHS.

This is a graph of a linear inequality:

Table of Contents Show

Linear Inequality
How to Graph a Linear Inequality
Example: y ≤ 2x−1
Example: 2y − x ≤ 6
Example: y/2 + 2 > x
Two Special Cases
What is the graph of linear inequality called?
What is the graph of a linear equation?
What is a linear inequality example?
What is the graphing solutions of a linear inequality?

The inequality y ≤ x + 2

We 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

Graph the "equals" line, then shade in the correct area.

Follow these 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

We can also have a horizontal, or vertical, line:

7426, 7427, 7428, 7429, 7430, 7431, 7432, 7433, 7434, 7435

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.