Graphical Solution of a System of Linear Inequalities

Introduction

Linear inequalities are numerical or algebraic expressions that compare two values using inequality symbols such as (less than), > (greater than), (less than or equal to), and (greater than or equal to) (not equal to). 

For example, numerical inequalities include 10 > 9, 20 > 17, whereas algebraic inequalities include x > y, y < 19 – x, x ≥ z > 11, and so on. Literal inequalities are another name for algebraic inequalities.

How to Solve Linear Inequalities Using Graph?

Graphing linear inequalities is a straightforward way to solve them. To draw the inequalities, repeat the procedures above. The coloured area, once drawn, is a solution to the inequity. If there are two or more inequalities then the common intersection of all inequalities in the region needed.

How to Draw Linear Inequalities On Graph?

The only difference which makes linear equality different from the linear equation is that the inequality sign replaces the’ = ‘sign. Graphing linear inequalities follow the same methods and concepts as graphing linear equations.

A linear equation produces a line graph while inequality produces a region, which is the only difference between the two equations. 

A boundary divides the coordinate plane into two sections in a linear inequality graph. All remedies to inequality are concentrated in one portion of the region. A dashed line indicating ‘>’ and a solid line representing ” >= ” which means borderline should also be included.

The steps for charting the system of linear inequalities are as follows:

• In an inequality equation, make the formula’s subject. For instance, y > x + 2
• Substitute an equal sign for the inequality sign, and choose arbitrary values for y and x.
• Create a line graph for these arbitrary x and y values.
• If the inequality sign is either ‘>=’ or ‘>’, remember to draw a solid line or a dashed line respectively, similarly for ‘<=’ and ‘<’.
• As we know the inequality ‘>’ or’ >=’ and ‘<’ or ‘<=’ divides the region in two parts, a shade above and below the line, correspondingly.

Let’s look at some examples to solve the following system of inequalities graphically.

Example 1

3y − x ≤ 6

Solution

Make y the subject of the formula to begin graphing this inequality.

When you add x to both sides, you get;

3y ≤ x + 6

Divide both sides by 2;

y ≤ x/3 + 2

Because of the sign, depict the equation y = x/3 + 2 as a solid line. Because of the indication, the shade is below the line.

Example 2

y/2 + 2 > x

Solution

Make y the formula’s subject.

Subtract 2 from both sides.

y/2 > x − 2

To get rid of the fraction, multiply both sides by two:

y > 2x − 4

Plot a dashed line of y = 2x 4 because of the > sign.

Example 3

Graph the following inequality to solve it: 2x – 3y ≥ 6

Solution

The first is to change the subject of the line 2x – 3y – 6 to y.

2x from both sides of the equation is subtracted.

2x – 2x – 3y ≥ 6 – 2x

-3y ≥ 6 – 2x

Reverse the sign by multiplying both sides by -3.

y ≤ 2x/3 -2

Create a graph with the equation y = 2x/3 – 2 and shade below the line.

Example 4

x + y < 1

Solution

Make y the subject of the expression by rewriting the equation x + y = 1. We’ll use a dotted line to build our graph because of the inequality sign.

Because of the indication, we shade above the dotted line after sketching it.

Example 5

How to solve the system of inequalities graphically?

y ≤ x

y ≥ -x

x = 5

Solution

Make a list of all the inequities.

Red represents y ≤ x

Blue represents y ≥ -x

Green represents line x = 5

The graphical solution to these inequalities is the common shaded region. 

For systems of equations, there are three possible outcomes.

Remember that the values that are true for all of the equations in a system of equations are called the solution. When solving a system of linear equations, there are three possible results. 

The graphs of equations within a system can tell you how many different solutions there are. 

• One Solution: When an ordered pair intersects a system of equations, the system has just one solution.

• Infinite Solutions: When the two equations graph as the same line, we have an infinite number of possible solutions.

• No Solution: There are no solutions when the lines that make up a system are parallel since they share no points in common.

Conclusion 

Linear inequalities are numerical or algebraic expressions that compare two values using inequality symbols such as (less than), > (greater than), (less than or equal to), and (greater than or equal to) (not equal to). A boundary divides the coordinate plane into two sections in a linear inequality graph. If there is more than one equality then common region will be the solution. While solving a line we can have three possible results:

• One Solution

• Infinite Solutions

• No Solution

Hopefully, you’ve understood how to solve the system of inequalities graphically!