5(x + 1) − x > 2x + 2 4(x + 1) − 2 ≤ 2(2x + 1) − x

Let's solve each inequality step by step:

1. 5(x + 1) − x > 2x + 2

First, distribute 5 to both terms inside the parentheses: 5x + 5 - x > 2x + 2

Combine like terms: 4x + 5 > 2x + 2

Next, isolate the variable on one side: 4x - 2x > 2 - 5

Combine like terms: 2x > -3

Finally, divide both sides by 2 to solve for x: x > -3/2

So the solution to the first inequality is x > -3/2.

2. 4(x + 1) - 2 ≤ 2(2x + 1) - x

First, distribute 4 and 2 to both terms inside the parentheses: 4x + 4 - 2 ≤ 4x + 2 - x

Combine like terms: 4x + 2 ≤ 3x + 2

Next, isolate the variable on one side: 4x - 3x ≤ 2 - 2

Combine like terms: x ≤ 0

So the solution to the second inequality is x ≤ 0.

Therefore, the combined solution for both inequalities is -3/2 < x ≤ 0.