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1). 6(x + 2) >  x - 3

Simplify both sides of inequalities:
6x + 12 > x - 3

Subtract x from both sides:
6x + 12 - x >  x - 3 - x

5x + 12 >  - 3


Subtract 12 from both sides:
5x + 12 - 12 > - 3 - 12
5x >  - 15

Divide both sides by 5:
5x/5 >  -15/5
Answer:  x > - 3


3). - 2(x + 6) <  x  - 20
Simplify both sides:
-2x - 12 < x - 20

Subtract x from both sides:
- 2x - 12 - x <  - 20 - x
- 3x - 12 <  - 20

Add 12 to both sides: 
- 3x - 12 + 12 <  - 20 + 12
- 3x <  - 8

Divide both sides by - 3:
- 3x/ - 3 <  - 8/ - 3

Answer:  x  > 8/3



4).  x - 9 <  3(x - 3)
Simplify both sides of inequalities:
x - 9 < 3x - 9


Subtract 3x from both sides:
x - 9 - 3x < 3x - 9 - 3x
- 2x - 9  <  - 9

Add 9 to both sides:
- 2x - 9 + 9 <  - 9 + 9
- 2x <  0

Divide both sides by - 2:
- 2x/ - 2  <  0/2

Answer:  x  >  0


2). 3 + 4x is less than or equal to 2(1 + 2x)

Simplify both of the inequalities:
4x + 3 is less than or equal to 4x + 2

Subtract 4x from both sides:
4x + 3 - 4x is less than or equal to 4x + 2 - 4x

3 is less than or equal to 2

Subtract 3 from both sides:

3 - 3 is less than or equal to 2 - 3
0  is less than or equal to  - 1


Answer:  There are no Solutions






Hope that helps!!!! ( Answer: Letter Choice (B), There are no solution)

Inequalities help us to compare two unequal expressions. The correct option is B.

What are inequalities?

Inequalities help us to compare two unequal expressions. Also, it helps us to compare the non-equal expressions so that an equation can be formed. It is mostly denoted by the symbol <, >, ≤, and ≥.

Solving the given inequalities,

A.)

6(x+2)>x-3

6x + 12 > x - 3

6x - x > -12 - 3

5x > -15

x > -15/5

x > -5

Thus, the given inequality has a solution.

B.)

3 + 4x ≤ 2(1+2x)

3 + 4x ≤ 2 + 4x

3 - 2 ≤ 4x - 4x

1 ≤ 0

Thus, the given inequality does not have any solution.

C.)

-2(x+6) < x - 20

-2x - 12 < x - 20

-2x - x < 12 - 20

-3x < -8

x > 8/3

x> 2.667

Thus, the given inequality has a solution.

D.)

x - 9 < 3(x - 3)

x - 9 < 3x - 9

x - 3x < 9 - 9

-2x < 0

x > 0

Thus, the given inequality has a solution.

Hence, the inequality that does not have any solution is 3 + 4x ≤ 2(1+2x).

Learn more about Inequality:

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