Ch-6Linear Inequalities

Submitted by-Lucky Choudhary

Submitted To – Mr. N.K Rai Sir

Date – 8th Sept. 2014

YES We Can !! YES We Can !! always translate a statement always translate a statement problem in the form of an equationproblem in the form of an equationBy Using InequalitiesInequalities ???

• i.e., using equations which have the following signs between L.H.S And R.H.S

• For eg :- 40x + 20y ≥ 120

Properties of Inequalities.

Essentially, all of the properties that you learned to solve linear equations apply to solving linear inequalities with the exception that if you multiply or divide by a negative you must reverse the inequality sign.

So to solve an inequality just do the same steps as with an equality to get the variable alone but if in the process you multiply or divide by a negative let it ring an alarm in your brain that says "Oh yeah, I have to turn the sign the other way to keep it true".

Example:

8462 xx- 4x - 4x

862 x + 6 +6

142 x -2 -2

Ring the alarm! We divided by a

negative!

7xWe turned the sign!

Types of inequalitiesSTRICT

• The ineqalities with < or > The ineqalities with < or > between the L.H.S & R.H.Sbetween the L.H.S & R.H.S

SLACK

• The ineqalities with ≤, or ≥ The ineqalities with ≤, or ≥ between the L.H.S & R.H.Sbetween the L.H.S & R.H.S

• The ineqalities having the The ineqalities having the degree 1degree 1

Eg :- 5x +2y > 10Eg :- 5x +2y > 10

• The ineqalities having the The ineqalities having the degree 2degree 2 Eg :- 5x^2 +2y > 10Eg :- 5x^2 +2y > 10

LINEAR QUADRATIC

Rules For Solving InequalitiesRules For Solving Inequalities

Solving Linear Inequalities On A Number Line

Q1 » » Solve and Show solution on Number Line 7x +3 < 5x +9 Solve and Show solution on Number Line 7x +3 < 5x +9

Sol » » » » 7x – 5x < 9 – 3 » 2x < 6 » x < 37x – 5x < 9 – 3 » 2x < 6 » x < 3

Point to note : -Point to note : - If the equality had been 7x +3 ≤ 5x +9 the the If the equality had been 7x +3 ≤ 5x +9 the the number line would had been like thisnumber line would had been like this

It is important to notice the open or closed interval , which has to be It is important to notice the open or closed interval , which has to be used according to the sign between the inequalityused according to the sign between the inequalityWhat about solving this one What about solving this one

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

2 ≤ 3x – 4 ≤ 5 2 ≤ 3x – 4 ≤ 5 View Here

Q1 » » Q1 » » Solve and Show solution on Number Line 2 ≤ 3x – 4 ≤ 5Solve and Show solution on Number Line 2 ≤ 3x – 4 ≤ 5 Sol » » Sol » »

. . 2 ≤ 3x – 4 ≤ 5 » 2 + 4 ≤ 3x ≤ 5 + 4 » 2 ≤ 3x – 4 ≤ 5 » 2 + 4 ≤ 3x ≤ 5 + 4 » » 6 ≤ 3x ≤ 9 » 2 ≤ x ≤ 3 » 6 ≤ 3x ≤ 9 » 2 ≤ x ≤ 3

-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Closed Interval ( )Closed Interval ( )Open Interval ( )Open Interval ( )

Introduction To Types Of Graphs

I

II

X - Axis

Y - Axis

Left half plane

Right half plane

OO

X - Axis

Y - AxisUpper half

plane

Lower half plane

OO

Sol. » Sol. » Steps to solving and graphing the InequalitySteps to solving and graphing the Inequality

Step 1 :- Assume that X + Y = 5 and find the following values

Step 2 :- Plot these points on graph

Step 3 :- Choose the half by taking some value for X or Y. if it satisfies the

inequality then shade that region as the Answer and if doesn’t then the other graph is the solution

Qs. » Qs. » Solve Graphically x +y < 5 Solve Graphically x +y < 5 X 0 5

Y 5 0

Graph of Equation x + y = 5

If the inequality has ≤ or ≥ sign then the the line of

equation is also the part of the graph otherwise it isn’t

Graph of Given InequalityGraph of Given Inequality

Sol. » Sol. » Steps to solving and graphing the InequalitySteps to solving and graphing the Inequality

Step 1 :- Assume that X + Y = 5 and find the following values

Step 2 :- Plot these points on graph

Step 3 :- Choose the half by taking some value for X or Y. if it satisfies the

inequality then shade that region as the Answer and if doesn’t then the other graph is the solution

Step 4 : - For x > 3 , if we follow the above

3 steps we get the graphs (green colour)And the solution of the given question will be the part of the graph commonto both the inequalities. (Dark Green Colour)

Qs. » Qs. » Solve Graphically x +y < 5 , x > 3 Solve Graphically x +y < 5 , x > 3 X 0 5

Y 5 0

Graph of Equation x + y = 5

0

-1

1

2

3

4

5

6

-1 -2 2 1 4 33 5 6Graph of Equation x = 3

Hope All Of You Like the

Presentation

Thank You