Question

If \( x \) is a number such that \( x^2 - 5x + 4<0 \) and \( x^2 - 3x + 2<0 \), which of the following can be the value of \( x \)?

• A. 3.5
• B. 3.0
• C. 2.4
• D. 1.6
• E. 0.8

Hint:

When solving inequalities, always check for the intersection of the solutions to each inequality.

Answer 1

The Correct Option is C

Step 1: Solve the first inequality \( x^2 - 5x + 4<0 \).
Factor the quadratic inequality: \[ (x - 4)(x - 1)<0 \] The solution to this inequality is \( 1<x<4 \).
Step 2: Solve the second inequality \( x^2 - 3x + 2<0 \).
Factor the quadratic inequality: \[ (x - 2)(x - 1)<0 \] The solution to this inequality is \( 1<x<2 \).
Step 3: Combine the two inequalities.
The solution to both inequalities is the intersection of \( 1<x<4 \) and \( 1<x<2 \), which gives \( 1<x<2 \). Hence, the value of \( x \) must be between 1 and 2. Therefore, the only option that satisfies this condition is \( x = 2.4 \).
Step 4: Conclusion.
The correct answer is (C).