Chapter 5.5, Problem 16CYU

To determine

Tosolve:The given equation

Answer to Problem 16CYU

  $z<-2$

  

Explanation of Solution

Given information:

  $6\left(-2z+5\right)<-19z+16$

Concept Used:

When two quantities are not equal, the comparison between them is often presented by certain symbols, this comparison is known as inequality and the symbols used for such comparison are known as inequality symbols.

There are four inequality symbols, $>$ , $<$ ,$\ge$ , $\le$ namely “greater than”, “less than”, “greater than or equal to” and “less than or equal to” respectively.

On a number line, integers are placed at an interval of one and value if integers increases as we move left to right on the number line.

Calculation:

  $\begin{array}{l}6\left(-2z+5\right)<-19z+16\\ \Rightarrow 6\left(-2z\right)+6·5<-19z+16\mathrm{...}\left(\text{Distributive property}\right)\\ \Rightarrow -12z+30<-19z+16\\ \Rightarrow -12z+30+19z<-19z+16+19z\mathrm{...}\left(\text{addition property of inequality}\right)\\ \Rightarrow 7z+30<16\\ \Rightarrow 7z+30-30<16-30\mathrm{...}\left(\text{subtraction property of inequality}\right)\\ \Rightarrow 7z<-14\\ \Rightarrow \frac{7z}{7}<\frac{-14}{7}\mathrm{...}\left(\text{division property of inequality}\right)\\ \Rightarrow z<-2\end{array}$

Graph:

In the inequality $z<-2$ we can say $z$ in smaller than $-2$ and can plot the same on number line-

We will make anopen dot on $-2$ as it is not included in the inequality and then move towards left as $z$ in smaller than $-2$ .