Question

What is a possible value for \(x\) in the expression below? $$ -6<\frac{8}{3} x<-\frac{1}{4} $$ (A) 8 (B) 1 (C) \(-2\) (D) \(-5\)

Short answer

The possible value for \(x\) is \(-2\) (option C).

Step-by-step solution

Understanding the Inequality

We need to solve the compound inequality \(-6 < \frac{8}{3} x < -\frac{1}{4}\). This inequality has to be solved in parts, breaking it into two separate inequalities: \(-6 < \frac{8}{3} x\) and \(\frac{8}{3} x < -\frac{1}{4}\).

Solve the First Inequality

For the first inequality \(-6 < \frac{8}{3} x\), we need to isolate \(x\). Multiply both sides by \(\frac{3}{8}\) to get \(x > \left(-6 \times \frac{3}{8}\right)\). Simplifying, \(x > -\frac{18}{8}\) or \(x > -\frac{9}{4}\).

Solve the Second Inequality

For the second inequality, \(\frac{8}{3} x < -\frac{1}{4}\), similarly multiply both sides by \(\frac{3}{8}\) to get \(x < \left(-\frac{1}{4} \times \frac{3}{8}\right)\). Simplifying, \(x < -\frac{3}{32}\).

Combine the Inequalities

Now we have two simple inequalities: \(x > -\frac{9}{4}\) and \(x < -\frac{3}{32}\). The solution is the intersection of these, meaning \(-\frac{9}{4} < x < -\frac{3}{32}\).

Check Possible Values

We now compare possible values (A, B, C, D) with the inequality \(-\frac{9}{4} < x < -\frac{3}{32}\). Calculate their approximations to make it easier: \(-\frac{9}{4} \approx -2.25\) and \(-\frac{3}{32} \approx -0.09375\). Only \(x = -2\) from option (C) satisfies the compound inequality.

Key concepts

Compound Inequalities

Compound inequalities involve two separate inequalities joined by the word 'and' or 'or.' In our problem, the expression \[-6 < \frac{8}{3} x < -\frac{1}{4}\]is a classic example of a compound inequality. What's unique about these inequalities is that they set boundaries for a variable, requiring it to satisfy both conditions simultaneously. The given problem is essentially saying, "find the values of \(x\) that are greater than \(-\frac{9}{4}\) and less than \(-\frac{3}{32}\)." To solve, you break it into two problems:

• Solve \(-6 < \frac{8}{3}x\) to find one boundary.
• Solve \(\frac{8}{3}x < -\frac{1}{4}\) for the other boundary.

By finding overlapping solutions of these two separate inequalities, you can discover the valid values for the variable \(x\). In this case, we are seeking the intersection of both solutions, where both conditions hold true. Working through these calculations refines what values of \(x\) make sense within this context.

Mathematics Problem Solving

Mathematics problem-solving is about thinking logically and methodically to interpret and solve equations—like those you encounter with inequalities. Beyond just finding a solution, it's about understanding the process behind the balance of equations.In the scenario provided, you begin by isolating the variable \(x\) in each part of the compound inequality. Through steps like multiplication or division, you adjust the equation to solve for \(x\), maintaining the equality. Simplifying can sometimes be tricky, especially if fractions or negative numbers are involved.The key is step-by-step transformation, applying inverse operations to gradually bring \(x\) by itself on one side of the inequality sign. Additionally, checking your solution by substituting back is crucial to verify accuracy. This structured approach in breaking down a problem makes equations like these easier to tackle.

PSAT Math

For students preparing for exams like the PSAT, understanding compound inequalities is essential. The PSAT tests your ability to solve various math problems and concepts under the constraints of time, necessitating both precision and speed. Facing compound inequalities on the PSAT, as per examples like the one provided, illustrates how you're required to analyze and accurately resolve issues simultaneously. These exercises test not just computational skills but also logical reasoning and the ability to interpret mathematical statements quickly. Understanding the methodology of creating boundaries and then finding intersections is a key skill in navigating these exams. Practice Makes Perfect:

• Regular practice with similar problems strengthens your ability to identify solutions faster.
• Understanding the fundamentals of mathematical operations aids you in recognizing shortcuts and establishing more efficient solving methods.

Ultimately, PSAT isn't just about getting the right answer—it's about mastering the techniques to arrive there confidently and efficiently.