Question

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

A) 6

Step-by-step solution

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: \(-2 > z > \frac{10}{21}\)

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: \(9\left(\frac{-10}{21}\right) - 18 = -\frac{90}{21} - 18\)

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