### Topics covered in this snack-sized chapter:

A linear inequality involves a linear expression by using any of the relational symbols such as <, >, ≤ or ≥.

x + 1 < 5
Note that the pointed arrow is always pointing to smaller number.

Inequalities can have one of the following symbols:
- > Stands for greater than.

- ≤ Stands for less than or equal to.

- ≥ Stands for greater than or equal to.

The < and > signs defines the sense of the inequality.
Two inequalities are said to have the same sense if the signs of inequality point in the same direction.

x + 3 > 2 and x + 1 > 0
Two inequalities are said to have the opposite sense if the signs of an inequality point in the opposite direction.

x - 4 < 0 and x > - 4
When we plot an inequality on a number line we can use an open hole or closed hole at that point.
An open hole means that the point is not included.
A closed hole means that the point is included.
Graph of X > 10

- Note that the circle or hole at X = 10 is open, the arrow is pointing to number greater than 10.

Graph of X ≥ 10

- Note that the circle or hole at X = 10 is closed, the arrow is pointing to a number greater than or equal to 10.

Graph of -3 > X or X < -3

- Note that the circle or hole at X = -3 is open, the arrow is pointing to a number less than -3.

Graph of – 3 ≥ X or X ≤ -3

- Note that the circle or hole at X = -3 is closed and arrow is pointing to numbers less than or equal to -3.

If a > b then

If a > b then

###### Multiplication Property: when c is positive:

If a > b and c > 0 then

###### Multiplication Property: when c is negative:

If a > b and c < 0 then

###### Division Property: when c is positive:

If a > b and c > 0 then

###### Division Property: when c is negative:

If a > b and c < 0 then
A system of linear equations consists of two or more linear inequalities:
The solution of a system of linear inequalities in two variables is any ordered pair that satisfies both of the linear inequalities.
We can find the solution graphically:
- On plotting the ordered pairs on graph that satisfy the inequalities in the system.

The graph is called the solution region for the system.
Graph showing x – y < 2
Find the ordered pairs that satisfy the inequality.

Replace the inequality symbol with an equal sign and graph the corresponding linear equation.
- If inequality is ≤ or ≥: use solid line

- If inequality is < or >: use dashed line

Choose an assumed point in one of the half-planes that is not on the line.
- Substitute the coordinates of the test point into the inequality.

- If it makes the statement true, shade the half-plane containing this test point.

- If it makes the statement false, shade the half-plane not containing this test point.

Solve the two variable inequality for 2y + 3x > 5.

Given: 2y + 3x > 5
Write the given equation in a slope intercept form:
2y + 3x = 5

Subtract 3x on both sides:
2y + 3x – 3x = 5 – 3x

2y = 5 – 3x

Now, divide 2 on both sides:
2y/2 = -3x/2 + 5/2

y = -1.5x + 2.5

To find the y-intercept, put x = 0,
y = 0 + 2.5

y = 2.5

The y-intercept is 2.5
To find x-intercept, put y = 0, we get:
0 = -1.5x + 2.5

Subtract 2.5 on both sides:
0 - 2.5 = -1.5x + 2.5 - 2.5

-2.5 = -1.5x

Divide 1.5 on both sides:
-2.5/1.5 = -1.5x/1.5

-1.67 = -x

x = 1.67

The x-intercept is 1.67.