I need how to solve and verify the solutions:

Question

1. r+7>-5
2. -35 +6n<7n
3. -14+p>4-(-2p)
4. x-5>-3/3
5. 4(n-1)<7n+8

Answer 1

1). r  + 7 > - 5

Subtract 7 from each side.

r  + 7 - 7 > - 5 - 7

r > - 12.

Solution set is {r | r > - 12}.

Check :

Substitute r = - 11 in r  + 7 > - 5.

- 11 + 7 > - 5

- 4 > - 5.

The above statement is true, so r > - 12 is a solution of r  + 7 > - 5.

2). - 35 + 6n < 7n

Add 35 to each side.

6n < 7n + 35

Subtract 7n from each side.

- n < 35

Multiply each side by negative one and flip the symbol.

n > - 35.

Solution set is {n | n > - 35}.

Check :

Substitute n = 1 in - 35 + 6n < 7n.

- 35 + 6(1) < 7(1)

- 29 < 7.

The above statement is true, so n > - 35 is a solution of - 35 + 6n < 7n.

3). - 14 + p > 4 - (- 2p)

- 14 + p > 4 + 2p

Add 14 to each side.

p > 4 + 2p + 14

Subtract 2p from each side.

- p > 18

Multiply each side by negative one and flip the symbol.

p < - 18.

Solution set is {p | p < - 18}.

Check :

Substitute p = - 20 in - 14 + p > 4 - (- 2p).

- 14 + (- 20) > 4 - (- 2(- 20))

(- 14 - 20) > (4 + 2(- 20))

- 34 > (4 - 40)

- 34 > - 36.

The above statement is true, so p < - 18 is a solution of - 14 + p > 4 - (- 2p).

4). x - 5 > - 3/3

x - 5 > - 1

Add 5 to each side.

x  > - 1 + 5

x  > 4.

Solution set is {x | x > 4}.

Check :

Substitute x = 5 in x - 5 > - 3/3.

5 - 5 > - 3/3

5 - 5 > - 1

0 > - 1.

The above statement is true, so x  > 4 is a solution of x - 5 > - 3/3.

5). 4(n - 1) < 7n + 8

Apply distributive property : a (b - c) = ab - ac.

4n - 4 < 7n + 8

Add 4 to each side.

4n < 7n + 8 + 4

4n < 7n + 12

Subtract 7n from each side.

- 3n < 12

Divide each side by negative three and flip the symbol.

n > - 12/3

n > - 4.

Solution set is {n | n > - 4}.

Check :

Substitute n = 1 in 4(n - 1) < 7n + 8.

4(1 - 1) < 7(1) + 8

0 < 15.

The statement is true, so n > - 4 is a solution of 4(n - 1) < 7n + 8.