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Graphing Linear Inequalities

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 Sub Topics Inequality is all about comparing one quantity with the other, a branch under the study of pre algebra. To sketch the graph of an inequality firstly graph the corresponding equation. In mathematics, it is represented by the symbols <, >, $\leq$ and $\geq$ and the expression is said to be of the form ax + b. Linear inequality of degree one can have any number of variables. When an inequality is multiplied or divided by a negative value the direction of the inequality changes. Inequalities are of many types, it can be numerical, literal, double, strict or slack.

Graphing Linear Inequalities in Two Variables

A solution of a system of linear inequality is an ordered pair that makes each inequality true. In this section we will only focus on graphing linear inequalities in two variables.
Just follow the simple steps discussed below:
1) On the same coordinate system graph each inequality.

2) Color the intersection of the graphs. The points in shaded region are the solutions of the system.

3) Now pick any random point in the shaded region and verify.

The coordinates will satisfy each inequality of the original system.
Given below is an example for better understanding the concept:

Example 1: Plot the graph for the inequality
5x - 4y $\leq$ 7

Solution:

Given : 5x - 4y $\leq$ 7

The above inequality can be expressed as y = $\frac{5x - 7}{4}$

When x = 2, y = $\frac{5x - 7}{4}$

y = $\frac{5(2) - 7}{4}$

y = $\frac{3}{4}$

y = 0.75

when x = -1,

y = $\frac{5(-1) - 7}{4}$

y = $\frac{-5- 7}{4}$

y = $\frac{-12}{4}$

y = -3

When x = 4

y = $\frac{5(4) - 7}{4}$

y = $\frac{20 - 7}{4}$

y = $\frac{13}{4}$

y = 3.25

Plot the graph

y = $\frac{5x - 7}{4}$

Step 2:
Check the solution for (0, 0)

5x - 4y $\leq$ 7

5(0) - 4(0) $\leq$ 7

0 $\leq$ 7

The above inequality is true.

Therefore the point (0, 0) is satisfying the given inequality.

Graph of 5x - 4y $\leq$ 7 is shown below.

The colored region is the solution for the given problem.

Example 2: Plot y $\geq$ x$^{2}$ - 3

Solution: Given : y $\geq$ x$^{2}$ - 3

Plot y = x$^{2}$ - 3,

the given problem is a parabola.

From the above graph,

Check for the point (0, 0) in the below equation.

y $\geq$ x$^{2}$ - 3

0 $\geq$  0 - 3

We see that the above expression holds true.

The solution lies in the shaded region.

Graphing Linear Inequalities on a Number Line

Real numbers and Integers can be represented on a number line. The point on line associated with each number is graph of the number. Isolation of a variable is done to solve the system of linear inequalities. The open part in a inequality symbol always faces the larger quantity.

Given below are some examples.
Example 1: Solve and graph the solution set : 2(3x+ 2)  > 4x+ 5

Solution:

Given: 2(3x + 2) > 4x + 5

6x + 4 > 4x + 5

2x >1

x > $\frac{1}{2}$

The graph is given below.

Example 2: Solve and graph  4 $\leq$ 5x + 3 $\leq$ 10

Solution:

Given:  4 $\leq$ 5x + 3 $\leq$ 10

Subtract 3 from both sides
4 - 3 $\leq$ 5x + 3 - 3 $\leq$ 10 - 3

Now combine the like terms.

1 $\leq$ 5x $\leq$ 7

To simplify x divide through out by 5.

$\frac{1}{5}$ $\leq$ x $\leq$ $\frac{7}{5}$

Graph below is the graph.

Graphing Linear Inequalities Word Problems

Given below is an example of graphing linear inequality word problem.
Example 1: David is selling laptops and desktops to make money for his vacation.
Laptops cost $\$$65 and desktops cost \$$50. He needs to make atleast$\900.
He is sure that he can sell more than 20 laptops. Graph the inequalities and shade the region.

Solution: Let x : The number of laptops sold.

Let y : The number of desktops sold.

The given problem can be expressed in mathematical form as

65x + 50y $\geq$ 900

David is also sure that he can sell atleast 20 laptops.

An inequality to represent this situation is
x > 20

We need to now find the x intercept as well as the y intercept.

Let y = 0, so that we get the x intercept.

65x + 50y = 900

65x + 50(0) = 900

x = $\frac{900}{65}$

x = 13.85

Now let x = 0, we will get the y intercept.

65x + 50y = 900

50y = 900

y = 18

Plug in (0, 0)

65x + 50y $\geq$  900

65(0) + 50 (0) $\geq$  900

0 $\geq$  900

As 0 is not greater than 900 do not shade the sign that contains (0,0).

Graph is given below. The dark grey region is the solution for the given problem: