Chapter 5.5, Problem 61S

To determine

To solve the inequality:

  $5.6z+1.5<2.5z-4.7$

And check the solution

Answer to Problem 61S

  $\left(-\infty ,-2\right)$

Explanation of Solution

Given:

  $5.6z+1.5<2.5z-4.7$

Concept Used:.

• To check the answer solution for the given inequality
1. First check the solution at end point by converting it into an equation.
2. If the solution is $x<m$, then pick any value of x less than m and check whether it satisfies the inequality.
3. If the solution is $x>m$, then pick any value of x greater than m and check whether it satisfies the inequality.

Properties of inequality:

1. $b<c\Rightarrow ab<ac,\text{ if }a>0$
2. $b<c\Rightarrow ab>ac,\text{ if }a<0$

Calculation:

To solve the inequality:

  $5.6z+1.5<2.5z-4.7$

Subtract both sides of the inequality by $2.5z$ , as

  $\begin{array}{l}5.6z+1.5<2.5z-4.7\\ \Rightarrow 5.6z-2.5z+1.5<2.5z-2.5z-4.7\\ \Rightarrow 3.1z+1.5<-4.7\end{array}$

Now subtracting $1.5$ on both sides as,

  $\begin{array}{l}3.1z+1.5-1.5<-4.7-1.5\\ \Rightarrow 3.1z<-6.2\end{array}$

Divide both sides of the remaining inequality by $3.1$ as,

  $\begin{array}{l}3.1z<-6.2\\ \Rightarrow \frac{3.1z}{3.1}<-\frac{6.2}{3.1}\\ \Rightarrow z<-2\end{array}$

Thus, the inequality is true for z less than $-2$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,-2\right)$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting z equals to $-2$ in right hand side

  $\begin{array}{c}3.1\left(-2\right)=-6.2\\ =\text{Right Hand}\end{array}$

Now put any value any value of z less than $-2$ , say $-10$ and check whether it satisfies the inequality, as

  $\begin{array}{l}3.1\left(-10\right)=-31\\ \text{And }-31<-6.2\end{array}$

Since, both conditions are satisfied thus solution of inequality is correct.