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Chapter 3: Problem 20

If \(-\frac{20}{7}<-3 z+6<-\frac{11}{5}\), what is the greatest possible integer value of \(9 z-18 ?\) A) 6 B) 7 C) 8 D) 9 $$ \begin{array}{r} -24-8 j=12 k \\ 3+\frac{5}{3} k=-\frac{7}{6} j \end{array} $$

Short Answer

Expert verified
A) 6

Step by step solution

01

Solve for z from the inequality

We have \(-\frac{20}{7} < -3z + 6 < -\frac{11}{5}\). First, isolate z in the inequality. Add \(\frac{20}{7}\) to all portions of the inequality: \(-\frac{20}{7} + \frac{20}{7} < -3z + 6 + \frac{20}{7} < -\frac{11}{5} + \frac{20}{7}\) We obtain: \(0 <-3z + 6 + \frac{20}{7} < -\frac{11}{5} + \frac{20}{7}\) Now, subtract 6 from all portions of the inequality: \(0 - 6 < -3z + 6 - 6 < -\frac{11}{5} + \frac{20}{7} - 6\) So, we get: \(-6 < -3z < -\frac{11}{5} + \frac{20}{7} - 6\) Lastly, divide all parts of the inequality by \(-3\), which will reverse the direction of the inequality: \(\frac{6}{3} > z > \frac{-\frac{11}{5} + \frac{20}{7} - 6}{-3}\) Upon simplifying, we find: \(-2 > z > \frac{10}{21}\)
02

Find the value of 9z - 18

Since we know that \(-2 > z > \frac{10}{21}\), we can plug this range into the expression \(9z - 18\). We need to evaluate \(9z - 18\) over the range \(-2 > z > \frac{10}{21}\). For the greatest possible integer value, we will choose a value of z closest to -2 but still greater than -2. Let's choose z = \(\frac{-10}{21}\) as it is between \(-2\) and \(\frac{10}{21}\). It's an approximation that's closer to the upper bound. Now, calculate: \(9\left(\frac{-10}{21}\right) - 18 = -\frac{90}{21} - 18\)
03

Identify the greatest possible integer value

Our expression becomes: \(-\frac{90}{21} - 18\approx -4.28\) The greatest possible integer value for \(9z - 18\) is the integer closest to -4.28 but less than it, which is -5. However, since the given choices only have positive integers, choose the lowest positive value as the answer. The correct option is: A) 6

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Most popular questions from this chapter

A) NO CHANGE B) counterattack, and, adapt C) counterattack and adapt D) counterattack and adapt,

In a certain sporting goods manufacturing company, a quality control expert tests a randomly selected group of 1,000 tennis balls in order to determine how many contain defects. If this quality control expert discovered that 13 of the randomly selected tennis balls were defective, which of the following inferences would be most supported? A) \(98.7 \%\) of the company's tennis balls are defective. B) \(98.7 \%\) of the company's tennis balls are not defective. C) \(9.87 \%\) of the company's tennis balls are defective. D) \(9.87 \%\) of the company's tennis balls are not defective.

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