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

To solve these inequalities, we'll simplify the expressions step by step.

1. 5(x + 1) − x > 2x + 2

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

Combine like terms on both sides of the inequality: 4x + 5 > 2x + 2

Next, subtract 2x from both sides of the inequality: 4x - 2x + 5 > 2

Simplify: 2x + 5 > 2

Next, subtract 5 from both sides of the inequality: 2x + 5 - 5 > 2 - 5

Simplify: 2x > -3

Finally, divide both sides of the inequality by 2 (since we're trying to isolate x): 2x/2 > -3/2

Simplify: x > -3/2

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

2. 4(x + 1) - 2 ≤ 2(2x + 1) - x

First, distribute the 4 and 2 to both sets of parentheses: 4x + 4 - 2 ≤ 4x + 2 - x

Simplify: 4x + 2 ≤ 3x + 2

Next, subtract 3x from both sides of the inequality: 4x - 3x + 2 ≤ 3x - 3x + 2

Simplify: x + 2 ≤ 2

Finally, subtract 2 from both sides of the inequality: x + 2 - 2 ≤ 2 - 2

Simplify: x ≤ 0

The solution to the inequality is x ≤ 0.