Find vertices of system of inequalities calculator

Graphing Inequalities Calculator solves the inequality and displays the corresponding graph. An inequality consists of the "greater than", "lesser than", or "not equal to" sign. It is used to compare two quantities.

What is Graphing Inequalities Calculator?

Graphing Inequalities Calculator is an online tool that helps to find the graph for a given inequality. Inequalities are used to represent an unequal relationship between two algebraic expressions. To use the graphing inequalities calculator, enter the inequality in the given input box.

Graphing Inequalities Calculator

How to Use Graphing Inequalities Calculator?

Please follow the steps below to find the graph of the given inequality using the online graphing inequalities calculator:

• Step 1: Go to Cuemath's online graphing inequalities calculator.
• Step 2: Enter the function in the given input box.
• Step 3: Click on the "Compute" button to find the graph for the given inequality.
• Step 4: Click on the "Reset" button to clear the field and enter new values.

How Does Graphing Inequalities Calculator Work?

Suppose we want to solve a linear inequality. The general form of a linear equation in two variables is given by ax + by = c. Now the same equation can be represented as an inequality by replacing the "=" sign with the required inequality sign. Given below are the steps that can be used to graph such an inequality.

• Step 1: Irrespective of the inequality sign, we will first solve the linear inequality in the same way as we solve a linear equation.
• Step 2: The variable y will be to the left-hand side (L.H.S) of the equation. The rest of the terms will be shifted to the right-hand side (R.H.S).
• Step 3: Identify certain test points. This implies that we have to substitute the value of x with certain numbers and find the corresponding value of y.
• Step 4: Now we draw a straight line that passes through these test points on a graph.
• Step 5: Take the inequality into consideration and shade the required region. Shade above the line for a "greater than" inequality and below the line for a "lesser than" inequality. The shaded region contains all coordinates that can solve the inequality.

Similar steps can be used to solve other inequalities such as quadratic, cubic, etc.

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Solved Examples on Graphing Inequalities Calculator

Example 1:

Compute the graph for 2x + 3y ≤ 7 and verify it using the graphing inequalities calculator.

Solution:

Solving the inequality as an equation

2x + 3y = 7.

y = -(2/3)x + 7/3

x = 2, y = 1.

x = 5, y = -1

x = -7, y = 7.

Draw a line passing through these points on the graph.

As we have a "lesser than" inequality, we shade the region below the line.

Example 2:

Compute the graph for y > -5x + 2 and verify it using the graphing inequalities calculator.

Solution:

Solving the inequality as an equation

y = -5x + 2

x = 0, y = 2.

x = 1, y = -3

x = -1, y = 7.

Draw a line passing through these points on the graph.

As we have a "greater than" inequality, we shade the region above the line.

Similarly, you can try the graphing inequalities calculator to compute the graph for the following inequalities:

• y ≤ 2x + 5
• y > 5x2 - 5

☛ Related Articles:

• Inequalities
• Linear Inequalities

☛ Math Calculators:

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.

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