Question

1. $-n + 7(4 + 3n) \geq 5 - 3n$

Explanation:

Step1: Expand the left - hand side

We use the distributive property \(a(b + c)=ab+ac\) to expand \(7(4 + 3n)\). So \(7\times4+7\times3n = 28 + 21n\). The inequality becomes \(-n+28 + 21n\geq5-3n\).
Combine like terms on the left - hand side: \((-n + 21n)+28\geq5-3n\), which simplifies to \(20n+28\geq5 - 3n\).

Step2: Move all terms with n to the left and constants to the right

Add \(3n\) to both sides of the inequality: \(20n+3n + 28\geq5-3n+3n\), which gives \(23n+28\geq5\).
Subtract 28 from both sides: \(23n+28 - 28\geq5 - 28\), so \(23n\geq- 23\).

Step3: Solve for n

Divide both sides of the inequality \(23n\geq - 23\) by 23. Since 23 is a positive number, the direction of the inequality sign remains the same. We get \(n\geq\frac{-23}{23}\), which simplifies to \(n\geq - 1\).

Answer:

The solution to the inequality \(-n + 7(4 + 3n)\geq5-3n\) is \(n\geq - 1\).