Inequalities may be written with the symbols Shown here:-

< | Less than |

> | Greater than |

≤ | Less than or equal to |

≥ | Greater than or equal to |

A range of values is often expressed on a number line.

**Two ranges are shown below:-**

____|_____|_____|_____|_____|_____|_____|_____|_____|_____

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

- Represents the set of all numbers between -4 and 0 excluding the endpoints -4 and 0 or -4< x < 0

_____|_____|_____|_____|_____|_____|_____|_____|_____|____

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

- Represents the set of all numbers bet6ween -4 and 0 excluding 3, or -1 < x ≤ 3

**Rules for working with inequalities:-**

- Trea t inequalities like equations when adding or subtracting terms or when multiplying/Dividing by a positive number on both sides of the inequality.
- Flip the inequality sign if you multiply or divide both sides of an equality by a negative number.,

EXAMPLE: Solve for x and represent the solution set on a number line.

ANS:- Multiply both sides by 4 12-x ≥ 8

Subtract 12 from both sides -x ≥ -4

Multiply both sides by -1 and flip x ≤ 4

The Inequality symbol

**EXAMPLE:-**

3/2x – ¾x < 7/4x -1

Ans:- 3/2x -3/4x-7/4x < -1 ( Subtract -7/4x from both sides )

3/4x – 7/4x <-1

-4/4x < -1

-x < -1 ( multiply both sides by -1 )

** Absolute Value:**L The absolute value of a number describes how far that number is away from 0 on a number line. The symbol of absolute value is |number|.

**STEPS TO SOLVE ABSOLUTE VALUE EQUATION:-**

- Take what’s the absolute value sign and set up two equations.First sets the positive value equal to the other side of the equation, and the second sites the negative value equal to the other side.
**Solve both Equations:**

EXAMPLE:- |2 X +4 | = 30

2X+4=30 OR -(2x+4)=30

X=13 +2x+4= -30

X= – 17

X=13 or (-17)

**Inequalities and absolute value:-**

STEPS FOR SOLVING QUESTIONS INVOLVING BOTH INEQUALITIES & ABSOLUTE VALUE:-

**Set up Two Equations:-**The first inequality replaces the absolute value with the positive of what’s inside and the second replaces the absolute value with the negative of what’s inside

EXAMPLE :- | x | ≥ 4

+( x ) ≥ 4 or –x ≥ 4

X ≥ 4 or x ≤ -4

_____|_____|_____|_____|_____|_____

-8 -4 0 4 8

Example :- | x+3 | ≤ 5

+( x + 3) < 5 and -( x+3 ) < 5

X < 2 x > -8

_____|____________________|______|_____

-8 0 2

The only numbers that make the original inequality true are those that are true for both inequalities. X should be greater than -8 and less than 2.

**NOTE:- In conclusion, for solving absolute value expressions that are greater than some quantity set up two equations & after sol;** having** two equations.We have to take real numbers that are satisfied by either equation. However, for solving absolute value expressions that are less than some quantity set up two equations that are less than some quantity, We have to take real numbers that are satisfied by both the equations.**

EXAMPLE :- If | 3x + 7 | ≥ 2x + 12 , then which of the following is TRUE?

- A) x ≤ -19/5 (B) x ≥ -19/5 (C) x ≥ 5 (D) x ≤ 19/5

**Ans:-** ( 3x + 7 ) ≥ 2x +12 -( 3x + 7 ) ≥ 2x + 12

x ≥ 5 -3x ≥ 2x +19

= -5x ≥ 19

=x ≤ -19/5

Using extreme values: Extreme values are helpful where the questions involve the potential range of value for variables in the problem.

QUES)- If -7 ≤ a ≤ 6 and -7 ≤ b ≤ 8, What is the maximum possible value for ab?

Ans:- Let us consider the different extreme value scenarios for a , b and ab.

a | b | Ab |

Min -7 | Min -7 | 49 |

Min -7 | Max 8 | -56 |

Max 6 | Min -7 | -42 |

Max 6 | Max 8 | 48 |

We can easily reckon that ab is maximized. When we take the negative extreme values for both a and b.

Therefore maximum value of ab = 49