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. |

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.

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:**

__2(3x + 2) > 4x + 5__

**Given:**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:**

__4 $\leq$ 5x + 3 $\leq$ 10__

**Given:**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.**

**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:**