Chapter 5.3, Problem 47PPS

To determine

To solve the inequality:

  $0.7\left(2m-5\right)\ge 21.7$

And check the solution.

Answer to Problem 47PPS

  $[18,\infty )$

Explanation of Solution

Given:

  $0.7\left(2m-5\right)\ge 21.7$

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

Calculation:

To solve the inequality:

  $0.7\left(2m-5\right)\ge 21.7$

First divide both sides of the inequality by $0.7$ , as

  $\begin{array}{l}0.7\left(2m-5\right)\ge 21.7\\ \Rightarrow \frac{0.7\left(2m-5\right)}{0.7}\ge \frac{21.7}{0.7}\\ \Rightarrow 2m-5\ge \frac{217}{10}\times \frac{10}{7}\\ \Rightarrow 2m-5\ge 31\end{array}$

Now adding $5$ on both sides as,

  $\begin{array}{l}2m-5+5\ge 31+5\\ \Rightarrow 2m\ge 36\end{array}$

Divide both sides of the remaining inequality by $2$ as,

  $\begin{array}{l}2m\ge 36\\ \Rightarrow \frac{2m}{2}\ge \frac{36}{2}\\ \Rightarrow m\ge 18\end{array}$

Thus, the inequality is true for m greater than or equal to $18$ , thus the solution of given inequality in interval form is:

  $[18,\infty )$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting m equals to $18$ in right hand side

  $\begin{array}{c}0.7\left[2\left(18\right)-5\right]=0.7\left(36-5\right)\\ =0.7\left(31\right)\\ =21.7\\ =\text{Right Hand}\end{array}$

Now put any value any value of m greater than $18$ , say 30 and check whether it satisfies the inequality, as

  $\begin{array}{c}0.7\left[2\left(30\right)-5\right]=0.7\left(60-5\right)\\ =0.7\left(55\right)\\ =38.5\text{ Which is greater than }21.7\end{array}$

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