Question

If
A = {x: 11x - 5 ≥ 7x + 3, x ∈ R} and,
B = {x: 18x - 9 ≥ 15 + 12x, x ∈ R},

Find the range of set A ∩ B and represent it on a number line. ​

Answer 1

Step-by-step explanation:

Given :-

A = {x: 11x - 5 ≥ 7x + 3, x ∈ R}

B = {x: 18x - 9 ≥ 15 + 12x, x ∈ R}

To find :-

Find the range of set A ∩ B and represent it on a number line ?

Solution :-

Given sets are :

A = {x: 11x - 5 ≥ 7x + 3, x ∈ R}

11x-5 ≥ 7x+3

=> 7x+3 ≤ 11x-5

On adding 5 on both sides

=> 7x+3+5≤ 11x-5+5

=> 7x + 8 ≤ 11x

On subtracting 7x both sides

=> 7x+8-7x ≤ 11x-7x

=>8 ≤ 4x

On dividing by 4 both sides

=> 8/4 ≤ 4x/4

=> 2 ≤ x

=> x ≥ 2

A = {The elements greater than or equal to 2 }-(1)

B = {x: 18x - 9 ≥ 15 + 12x, x ∈ R}

18x-9≥ 15+12x

=> 15+12x ≤ 18x -9

on adding 9 both sides

=> 15+12x+9 ≤ 18x-9+9

=> 24+12x ≤ 18x

On Subtracting 12x both sides

=> 24+12x-12x ≤ 18x -12x

=> 24 ≤ 6x

On dividing by 6 both sides then

=> 24/6 ≤ 6x/6

=> 4 ≤ x

=> x ≥ 4

B = { The elements greater than or equal to 4 } -(2)

A ∩ B =

{The elements greater than or equal to 2 }∩ {The elements greater than or equal to 4 }

=> { x: x ≥ 2 , x ∈ R} ∩ { x: x ≥ 4 , x ∈ R}

=> {The elements greater than or equal to 4 }

A ∩ B = { x: x ≥ 4 , x ∈ R}

Solution :-

The possible values of A ∩ B are the real numbers greater than or equal to 4.

The range of A ∩ B = { x: x ≥ 4 , x ∈ R}

Range set of A ∩ B is on the number line

(see the attachment)

The range of the required set is on the right side to 4 on the number line (inclusive 4)

Used formulae :-

• Every number can be represented on the number line.

• This number line is called Real number line.

• The set of the common elements in both A and B is called the intersection of A and B sets and it is denoted by A ∩ B.

• The collection of rational numbers and irrational numbers is called Real and they are denoted by R

• If an element belongs to any set then we use the symbol ∈.

Answer 2

Answer:

See the attachment for answer:-

Since x is directly related to 'R' real numbers hence you need to draw a line in between two numbers. The arrow represents that solution is till infinity.

Note: As prescribed by our teachers in exams we have to draw the number line with pencil.Here I used pen as pencil will be not visible. Also, note that draw the number line using a ruler with correct measurement (1cm each).

The solutions are to be represented above the number line. If number is included, shade it otherwise just draw a circle. I don't recommend you to represent the answer in the number line as the examiner may be confused and may not give you marks. Hence, represent it properly so that there is no confusion.

Hope it helps you.