Chapter 5.4, Problem 31PPS

To determine

Define a variable, write an inequality and solve the problem and check the solution.

Answer to Problem 31PPS

Variable is the number: x

Compound Inequality: $35<-5x<10$

  $-7>x>-2$

Explanation of Solution

Given:

The product of − 5 and a number is greater than 35 or less than 10.

Concept Used:

Translate the statement into inequality equation.

Let the number be x.

The product of − 5 and a number: $-5x$

The product of − 5 and a number is greater than 35: $-5x>35$

Eight less than a number is no more than 14: $x-8\le 14$

The product of − 5 and a number less than 10: $-5x<10$

Compound Inequality: $35<-5x<10$

Calculation:

Compound Inequality: $35<-5x<10$

  $\begin{array}{l}35<-5x<10\\ \frac{35}{-5}>\frac{-5x}{-5}>\frac{10}{-5}[\text{Multiply}\text{both}\text{sides}\text{by}-5\text{and}\text{reverse}\text{the}\text{inequality}\text{]}\\ -7>x>-2\end{array}$

  $\begin{array}{l}\text{Check}\text{the}\text{solution}\text{when}x=-1\\ 35<-5x<10\\ 35<-5x<10\\ 35<-5·-1<10\\ 35<5<10[\text{True}\text{statement}]\end{array}$

The solution of the compound inequality $35<-5x<10$ is $-7>x>-2$

Thus, the solution of the compound inequality $35<-5x<10$ is $-7>x>-2$