Chapter 5.4, Problem 26PPS

To determine

To solve below compound inequality:

  $10m-7<17m,\text{ or }-6m>36$

And to graph the solution on a number line.

Answer to Problem 26PPS

No solution.

Explanation of Solution

Given:

  $10m-7<17m,\text{ or }-6m>36$

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. $b\le c\Rightarrow b\pm a\le c\pm a$
2. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$
3. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

In order to solve the compound inequality:

  $10m-7<17m,\text{ or }-6m>36$

First, simplify the inequality,

  $10m-7<17m$

Simplify it using properties of inequalities, as

  $\begin{array}{c}10m-7<17m\\ \Rightarrow 10m-7-10m<17m-10m\\ \Rightarrow -7<7m\\ \Rightarrow -\frac{7}{7}<\frac{7m}{7}\\ \Rightarrow -1<m\end{array}$

Now, simplify the inequality

  $-6m>36$

For that divide both sides of the inequality by $-6$ , that would change the inequality, as

  $\begin{array}{c}-6m>36\\ \Rightarrow -\frac{6m}{-6}<\frac{36}{-6}\\ \Rightarrow m<-6\end{array}$

Combining both of the above results, as

  $-1<m,\text{ and }m<-6$

But there does not exist any number that is greater than $-1$ but is smaller than $-6$ .

Thus, the given inequality has no solution.