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# Graphing Linear Inequalities – Explanation & Examples

Linear inequalities are numerical or algebraic expressions in which two values are compared by the use of inequality symbols such, < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), and ≠ (not equal to)

For example, 10 < 11, 20 > 17 are examples of numerical inequalities whereas, x > y, y < 19 – x, x ≥ z > 11 etc. are all the examples of algebraic inequalities. Algebraic inequalities are sometimes referred to as literal inequalities.

The inequality symbols ‘< ‘and ‘>’ are used to express the strict inequalities, whereas the symbols ‘≤’ and ‘≥’ represent slack inequalities.

## How to Graph Linear Inequalities?

A **linear inequality** is the same as a linear equation, only that the inequality sign substitutes the equals sign. The same steps and concepts used to graph linear equations are also applied to graph linear inequalities.

The only **difference between the two equations** is that a linear equation gives a line graph. In contrast, a linear inequality shows the area of the coordinate plane that satisfies the inequality.

A linear inequality graph usually uses a borderline to divide the coordinate plane into two regions. One part of the region consists of all solutions to inequality. The borderline is drawn with a dashed line representing ‘>’ and ‘<’ and a solid line representing ‘≥’ and ‘≤’.

*The following are the steps for graphing an inequality:*

- Given an inequality equation, make y the subject of the formula. For example, y > x + 2
- Substitute the inequality sign with an equal sign and choose arbitrary values for either y or x.
- Plot and a line graph for these arbitrary values of x and y.
- Remember to draw a solid line if the inequality symbol is either ≤ or ≥ and a dashed line for < or >.
- Do the shading above and below the line if the inequality is > or ≥ and < or ≤ respectively.

## How to Solve Linear Inequalities by Graphing?

Solving linear inequalities by graphing is really simple. Follow the above steps to draw the inequalities. Once drawn, the shaded area is a solution to that inequality. If there is more than one inequality, then the common shaded area is a solution to inequalities.

*Let’s understand this concept with the help of the examples below.*

*Example 1*

2y − x ≤ 6

__Solution__

To graph, this inequality, start by making y the subject of the formula.

Adding x to both sides gives;

2y ≤ x + 6

Divide both sides by 2;

y ≤ x/2 + 3

Now plot the equation of y = x/2 + 3 as a solid line because of the ≤ sign. The shade below the line because of the ≤ sign.

*Example 2*

y/2 + 2 > x

__Solution__

Make y the subject of the formula.

Subtract both sides by 2;

y/2 > x − 2

Multiply both sides by 2 to eliminate the fraction:

y > 2x − 4

Now, because of the > sign, plot a dashed line of y = 2x − 4.

*Example 3*

Solve the following inequality by graphing: 2x – 3y ≥ 6

__Solution__

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

Subtract 2x from both sides of the equation.

2x – 2x – 3y ≥ 6 – 2x

-3y ≥ 6 – 2x

Divide both sides by -3 and reverse the sign.

y ≤ 2x/3 -2

Now draw a graph of y = 2x/3 – 2 and shade below the line.

*Example 4*

x + y < 1

__Solution__

Rewrite the equation x + y = 1 to make y the subject of the formula. Because the inequality sign is <, we will draw our graph with a dotted line.

After drawing the dotted line, we shade above the line because of the < sign.

*Example 5*

Find the graphical solution of the following inequalities:

y ≤ x

y ≥ -x

x = 5

__Solution__

Draw all the inequalities.

Red represents y ≤ x

Blue represents y ≥ -x

Green represents line x = 5

The common shaded area (can be seen clearly) is the graphical solution to these inequalities.

** **

*Practice Questions*

1. Graph the solution to y < 2x + 3

2. Graph the inequality: 4(x + y) – 5(2x + y) < 6 and answer the questions below.

a. Check whether the point (-22, 10) is within the solution set.

b. Determine the slope of the border line.

3. Graph the inequality of y< 3x and determine which quadrant will be completely shaded.

4. Graph the inequality y > 3x + 1 and answer the questions below:

a. Is the point (-5, -2) within the solution set?

b. Is the borderline drawn dashed or solid? Explain your answer.

5. Draw a graph of 4x – 3y > 9 and answer the question below:

a. Determine whether the point (2, -2) is within the solution set.

b. Which quadrant has no solutions to this inequality?

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