Chapter 5.4, Problem 67S

To determine

To solve the inequality:

  $-3\left(7n+3\right)<6n$

And to graph the solution on a number line.

Answer to Problem 67S

  $\left(-\frac{1}{3},\infty \right)$

  

Explanation of Solution

Given:

  $-3\left(7n+3\right)<6n$

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Properties of inequality:
1. $b<c\Rightarrow b\pm a<c\pm a$
2. $b<c\Rightarrow ab<ac,\text{ if }a>0$
3. $b<c\Rightarrow ab>ac,\text{ if }a<0$

Calculation:

In order to solve the inequality

  $-3\left(7n+3\right)<6n$

First, simplify it as

  $\begin{array}{c}-3\left(7n+3\right)<6n\\ \Rightarrow -21n-9<6n\end{array}$

Now, add $21n$ both sides of the inequality and then further simplify it using properties of the inequalities, as

  $\begin{array}{c}-21n-9<6n\\ \Rightarrow -21n-9+21n<6n+21n\\ \Rightarrow -9<27n\\ \Rightarrow -\frac{9}{27}<\frac{27n}{27}\\ \Rightarrow -\frac{1}{3}<n\end{array}$

Thus, the solution of given inequality is every number strictly greater than $-\frac{1}{3}$ , this can be represented in interval form, as

  $\left(-\frac{1}{3},\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-\frac{1}{3}$ , is strict, so use a hollow circle at $-\frac{1}{3}$ . And the solution interval has $\infty$ at right endpoint which means every number strictly greater than $-\frac{1}{3}$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: