Chapter 3.4, Problem 7P

Solve each inequality.

$2(k+4)-3k\le 14$

To determine

ToSolve:The inequality $2\left(k+4\right)-3k\le 14$

Answer to Problem 7P

The set of solutions of the given inequality are all the values the interval $\left[-6,\infty \right)$

Explanation of Solution

Given:

The inequality $2\left(k+4\right)-3k\le 14$

Concept Used:

The solution of aninequality is the set of all the values which satisfies the inequality.

For any three numbers $a,b$ and $c$

  $a\left(b+c\right)=ab+ac$

Calculation:

Given the inequality $2\left(k+4\right)-3k\le 14$

Solving it, we have

  $\begin{array}{c}2\left(k+4\right)-3k\le 14\\ 2k+\left(2\right)\left(4\right)-3k\le 14\text{ Using }a\left(b+c\right)=ab+ac\\ 2k+8-3k\le 14\\ -k+8\le 14\end{array}$

  $\begin{array}{c}-k+8-8\le 14-8\text{ Subtract 8 from both sides}\\ -k\le 6\\ \left(-1\right)\left(-k\right)\ge \left(-1\right)\left(6\right)\text{ Multiply by }-1\text{ to both sides}\\ k\ge -6\end{array}$

That is $k\in \left[-6,\infty \right)$

Therefore, the set of solutions of the given inequality are all the values the interval $\left[-6,\infty \right)$

Conclusion:

The set of solutions of the given inequality are all the values the interval $\left[-6,\infty \right)$