Chapter 3.5, Problem 1STP

For Exercises $1-6,$ choose the correct letter.

What is the solution of $-(3n-7)\ge 4$ written in set-builder notation?

A. $n\le 1$

B. $\left\{1\right\}$

c. $\{n\mid n\ge 1\}$

D. $\{n\mid n\le 1\}$

To determine

To solve the inequality $-\left(3n-7\right)\ge 4$ and to write the solution in set-builder notation.

Answer to Problem 1STP

Solution is $\left\{n/n\le 1\right\}$ and option D is correct.

Explanation of Solution

Given:

  $-\left(3n-7\right)\ge 4$

Concept used:

When multiplying or dividing each side of an inequality by the same negative number, the direction of the inequality symbol must be reversed to produce an equivalent inequality.

Calculation:

  $\begin{array}{l}-\left(3n-7\right)\ge 4\\ \Rightarrow -3n+7\ge 4\end{array}$

Subtracting 7 on both sides, we get

  $\begin{array}{l}-3n+7-7\ge 4-7\\ \Rightarrow -3n\ge -3\end{array}$

Dividing by $-3$ on both sides, we get

  $\begin{array}{l}\frac{-3n}{-3}\le \frac{-3}{-3}\\ \Rightarrow n\le 1\end{array}$

Therefore, set-builder form is,

  $\left\{n/n\le 1\right\}$

Hence option D is correct.