# Chapter 3 : Linear Inequalities

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

#### Linear Inequalities arrow_upward

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

• ###### Example:

• x + 1 < 5
• Note that the pointed arrow is always pointing to smaller number.
• • Inequalities can have one of the following symbols:
• < Stands for less than.
• > 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.

• ###### Example:

• 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.

• ###### Example:

• x - 4 < 0 and x > - 4

• #### Graphical Representation of Inequalities arrow_upward

• 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.

#### Properties Used to Solve Linear Inequalities arrow_upward

• If a > b then
• a + c > b + c

###### Subtraction Property:

• If a > b then
• a – c > b – c

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

• If a > b and c > 0 then
•  a × c > b × c

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

• If a > b and c < 0 then
• a × c < b × c

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

• If a > b and c > 0 then
• a/c > b/c

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

• If a > b and c < 0 then
• a/c < b/c

#### System of Linear Inequalities arrow_upward

• A system of linear equations consists of two or more linear inequalities:
• 2x + 2y > 12
• 2x - 2y > 0
• 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.

• #### Graphing Inequalities in Two Variables arrow_upward

• 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.

#### Solving Inequalities in Two Variables arrow_upward

###### Example:

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

• ###### Solution:

• 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.

• #### Thank You from Kimavi arrow_upward

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