The values of p for which the expression `(px^(2)+3x-4)/(p+3x-4x^(2))` can assume real values for real x lie in the interval

Options

A. `p le 1" or "p ge 7`
B. `p ge 1" or "p le 7`
C. `1 lt p lt 7`
D. none of these

Answer

To find the values of \( p \) for which the expression

\[ \frac{px^2 + 3x - 4}{p + 3x - 4x^2} \]

can assume real values for real \( x \), we need to ensure that the denominator does not equal zero and that the overall expression is defined for all real \( x \).

### Step 1: Identify the denominator
The denominator is given by:

\[ p + 3x - 4x^2 \]

For the expression to be defined for all real \( x \), the denominator must not be zero for any real \( x \). This means we need to analyze the quadratic equation:

\[ -4x^2 + 3x + p = 0 \]

### Step 2: Find the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

For our equation, \( a = -4 \), \( b = 3 \), and \( c = p \). Thus, the discriminant becomes:

\[ D = 3^2 - 4(-4)(p) = 9 + 16p \]

### Step 3: Set the discriminant condition
For the quadratic to have no real roots (which means the denominator does not equal zero for any real \( x \)), the discriminant must be less than zero:

\[ 9 + 16p < 0 \]

### Step 4: Solve the inequality
Now, we solve the inequality:

\[ 16p < -9 \]
\[ p < -\frac{9}{16} \]

### Step 5: Analyze the numerator
Next, we also need to ensure that the numerator \( px^2 + 3x - 4 \) is defined for all \( x \). The quadratic \( px^2 + 3x - 4 \) can assume real values for all \( x \) if its discriminant is non-negative:

\[ D' = 3^2 - 4p(-4) = 9 + 16p \]

For this quadratic to have real roots, we need:

\[ 9 + 16p \geq 0 \]

### Step 6: Solve the second inequality
Solving this gives:

\[ 16p \geq -9 \]
\[ p \geq -\frac{9}{16} \]

### Step 7: Combine the conditions
Now we have two conditions:

1. \( p < -\frac{9}{16} \)
2. \( p \geq -\frac{9}{16} \)

These two conditions cannot be satisfied simultaneously. Therefore, there are no values of \( p \) for which the expression can assume real values for all real \( x \).

### Conclusion
The values of \( p \) for which the expression can assume real values for real \( x \) lie in the interval:

\[ \text{No valid interval} \]