Question

If \(x^2 - 5x + 6<0\), then which of the following is a possible value of x?

• A. 1
• B. 2
• C. 3
• D. 4
• E. 2.5

Hint:

For a quadratic inequality of the form \((x-a)(x-b)<0\) with \(a<b\), the solution is always the interval between the roots: \(a<x<b\). For \((x-a)(x-b)>0\), the solution is outside the roots: \(x<a\) or \(x>b\). Recognizing this pattern can save you time. Always be careful with strict inequalities (<, >) versus inclusive inequalities ($\leq$, $\geq$).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question requires solving a quadratic inequality. The goal is to find the range of values for \(x\) that make the expression \(x^2 - 5x + 6\) negative.
Step 2: Key Formula or Approach:
To solve a quadratic inequality, we first find the roots of the corresponding quadratic equation (\(ax^2 + bx + c = 0\)). These roots are the critical points that divide the number line into intervals. We then test a value from each interval to see if it satisfies the inequality.
Step 3: Detailed Explanation:
1. Find the roots of the equation:
Set the expression equal to zero: \[ x^2 - 5x + 6 = 0 \] Factor the quadratic expression. We need two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. \[ (x - 2)(x - 3) = 0 \] The roots are \(x = 2\) and \(x = 3\).
2. Analyze the inequality:
The inequality is \((x - 2)(x - 3)<0\). This means the product of the two factors must be negative. This occurs when one factor is positive and the other is negative.

Case 1: \(x - 2>0\) and \(x - 3<0\)
This implies \(x>2\) and \(x<3\). Combining these gives the interval \(2<x<3\).

Case 2: \(x - 2<0\) and \(x - 3>0\)
This implies \(x<2\) and \(x>3\). It is impossible for \(x\) to be both less than 2 and greater than 3. So, this case yields no solution.

The solution to the inequality is \(2<x<3\). The value of \(x\) must be strictly between 2 and 3.
3. Check the options:
Now, let's examine the given integer options:
(A) 1: Is not between 2 and 3.
(B) 2: Is not strictly greater than 2. If \(x=2\), the expression equals 0, but the inequality is strictly less than 0 (\(0<0\) is false).
(C) 3: Is not strictly less than 3. If \(x=3\), the expression equals 0 (\(0<0\) is false).
(D) 4: Is not between 2 and 3.
Step 4: Final Answer:
The solution to the inequality \(x^2 - 5x + 6<0\) is the set of all numbers \(x\) such that \(2<x<3\). None of the integer options provided (1, 2, 3, 4) fall within this range. Therefore, there is likely an error in the question or the options provided, as none of them are correct. A possible value for \(x\) would be 2.5, but this is not an option.