Chapter 8, Problem 58CR

To determine

Solve the inequality $\frac{2}{3}k+9<5$ and check your solutions

Answer to Problem 58CR

  $k<-6$

Explanation of Solution

Given:

The inequality, $\frac{2}{3}k+9<5$

Concept used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• Inequality sign get reverse on multiplying or dividing by a negative number

Calculation:

In order to solve the inequality $\frac{2}{3}k+9<5$ , first isolate the variable term on left side by performing some basic operations. Here to isolate k, first subtract $9$ both sides and then multiply both sides by $5$ and then divide both sides by $2$ and simplify further to get

  $\begin{array}{c}\frac{2}{3}k+9<5\\ \frac{2}{3}k+9-9<5-9\\ \frac{2}{3}k<-4\\ \frac{2}{3}k\times 3<-4\times 3\\ 2k<-12\\ \frac{2k}{2}<\frac{-12}{2}\\ k<-6\end{array}$

Thus, the solution of the given inequality is $k<-6$ .

Now in order to check the solution of the equation, substitute $k<-6$ in the given equation and check whether the left hand side and right hand side are equal. So, it gives

  $\begin{array}{c}\frac{2}{3}\left(-3\right)+9<5\\ -2+9<5\\ -2+9<5\\ 7<5False\text{ statement}\end{array}$

Thus, the solution of the given equation is not $k<-6$ .