Question

If \(x>2\) and \(x<3\), then which of the following is positive? (I) \(\quad(x-2)(x-3)\) (II) \(\quad(2-x)(x-3)\) (III) \((2-x)(3-x)\) (A) I only (B) II only (C) III only (D) I and II only (E) I and III only

Short answer

(B) II only.

Step-by-step solution

Analysis of Range for x

We know that \(x\) is within the range \(2 < x < 3\). Let's analyze how this affects each expression separately. We will substitute sample values for \(x\) from within this range and check whether the expression is positive or negative.

Expression (I) Analysis

Consider \((x-2)(x-3)\). Choosing a test value, say \(x = 2.5\):Substitute into the expression:\[(x-2)(x-3) = (2.5-2)(2.5-3) = (0.5)(-0.5) = -0.25\]Since the test result is negative, expression (I) is not positive.

Expression (II) Analysis

Consider \((2-x)(x-3)\). Again, using the same test value \(x = 2.5\):Substitute into the expression:\[(2-x)(x-3) = (2-2.5)(2.5-3) = (-0.5)(-0.5) = 0.25\]Since the result is positive, expression (II) is positive for \(x\) in this range.

Expression (III) Analysis

Consider \((2-x)(3-x)\). Using the same test value \(x = 2.5\):Substitute into the expression:\[(2-x)(3-x) = (2-2.5)(3-2.5) = (-0.5)(0.5) = -0.25\]Since the result is negative, expression (III) is not positive.

Conclusion

From the analysis, only expression (II) is positive when \(x\) is between 2 and 3. Therefore, the correct answer is (B) II only.

Key concepts

Inequalities

Understanding inequalities is crucial for solving many types of math problems, especially those involving ranges like this one. Inequalities show that one value is less than, greater than, not equal to, etc., another value. In this exercise, we were given inequalities of the form \(2 < x < 3\), meaning that \(x\) can be any value between 2 and 3 but not including 2 or 3.

To solve problems involving inequalities, it's often helpful to select test values within the specified range, which helps determine the sign (positive or negative) of mathematical expressions you need to evaluate. Selecting clean numbers like 2.5 from the range makes calculations simple and reliable. Always remember the purpose is to see what happens to expressions as \(x\) changes within the constraints given by the inequality.

When working with inequalities related to real-world applications or complex problems, remember they simply express relationships of size and magnitude within set bounds. This understanding allows for evaluating circumstances and data more flexibly and accurately.

Expression Analysis

Expression analysis involves breaking down algebraic expressions to understand them better. In this case, expressions are products of linear factors, which we have evaluated. Consider expression (I): \((x-2)(x-3)\). When you break this down, you are essentially looking at two factors: \(x-2\) and \(x-3\).

Each factor behaves differently in different parts of the range \(2 < x < 3\). For instance, \(x-2\) will be positive for any \(x > 2\), while \(x-3\) will always be negative in our range because \(x\) does not reach 3. Multiplying a positive number with a negative one results in a negative value. That's why expression (I) yields a negative product here.

In expression (II), \((2-x)(x-3)\), both factors \(2-x\) and \(x-3\) become negative in our specific range, resulting in a positive product when multiplied together because multiplying two negative numbers results in a positive number. Analyzing expressions in this manner helps identify how the interaction between terms determines the overall value.

Range of Values

The range of values refers to the interval within which our variable, \(x\), must lie. In this problem, \(x\) is confined to values greater than 2 but less than 3, defined as \(2 < x < 3\).

Identifying the range of values helps in determining how expressions behave, as was seen with our expressions. The behavior of each factor within an expression is ultimately reliant on its interaction within this range. That's why substituting a value within this range, like \(x = 2.5\), is common practice to easily confirm the positivity or negativity of an expression.

Understanding the range of values is especially useful when identifying valid solutions or conditions in broader contexts. For equations or inequalities you might face in the GMAT or real-world problems, clearly defining your range or domain is a key step to ensuring correct solutions.