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

Key concepts

Inequalities

Inequalities are vital in SAT math problems because they allow us to compare sizes and limits of values, ensuring students grasp fundamental concepts. When faced with an inequality like the one in the exercise, the primary goal is to identify the range of possible values for a variable. In our problem, we're working with the inequality:

• -\(\frac{20}{7} < -3z + 6 < -\frac{11}{5} \)

To simplify, isolate the variable by manipulating the inequality through basic algebraic operations like addition, subtraction, multiplication, or division.
For example, to get the variable z alone, we first must add or subtract any constants affecting it, and then divide or multiply to adjust the coefficient of the variable. It's important to note that if you multiply or divide by a negative number, the inequality's direction reverses. This reversal highlights a key distinctive feature of working with inequalities.
Mastering inequalities prepares you to solve more complex algebraic problems on the SAT.

Integer Solutions

Integer solutions are critical when solving SAT math problems as they often require you to find whole number answers, reflecting quantities in real-world scenarios. In our problem, determining the greatest possible integer value of the expression \(9z-18\) involves finding valid whole numbers that satisfy given conditions.
When the solution describes the range

• \(-2 > z > \frac{10}{21}\)

we need to recognize that although z is a fraction here, the final step involves determining how integer answers affect the result. This might not directly give you whole numbers for z, but plugging integers into expressions like \(9z-18\) will. Generally, this challenge involves testing integer inputs within a range to find the best possible fit.
Understand this approach ensures you are prepared when integer solutions are explicitly asked in questions and helps develop problem-solving skills to tackle similar SAT questions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent mathematical relationships and are abundant in SAT exams. In the exercise, simplifying the expression \(9z-18\) is crucial to finding the solution.

• Break down the expression by substituting possible values of variables derived from inequalities.
• Perform operations like multiplication and subtraction as required, and then evaluate them.
  

Understanding these steps guarantees that you can decode any algebraic puzzle and find logical outcomes based on variable relationships. Always watch out for simple arithmetic errors or order of operations mistakes, which could lead to incorrect results.
Practicing these techniques will build your confidence in managing algebraic expressions and prepare you well for the SAT.

SAT Preparation

Effective SAT preparation is critical for mastering concepts like inequalities, integer solutions, and algebraic expressions. The SAT tests not only your mathematical skills but also your logical reasoning and problem-solving abilities.
To prepare for this:

• Engage with practice problems similar to the given exercise to understand typical SAT patterns.
• Review core algebraic concepts regularly to build a strong foundation.
• Take timed practice tests to improve your time management skills during the real exam.

Additionally, tackle problems with a strategic approach: understand the problem, plan your solution path, execute your plan, and verify your answer. This method ensures accuracy and boosts confidence.
With consistent practice and strategic studying, you'll be well-equipped to handle any SAT challenge that comes your way.