Find the value of a such that `-a.9^(x)+(a-2)3^(x)-((5)/(4)a-1)gt 0` for `x in (0,1)`. Where `a < 0`.

Answer

To solve the inequality \(-a \cdot 9^x + (a - 2) \cdot 3^x - \left(\frac{5}{4}a - 1\right) > 0\) for \(x \in (0, 1)\) and \(a < 0\), we will follow these steps:

### Step 1: Substitute \(3^x\) with \(y\)
Let \(y = 3^x\). Then, we have:
\[ 9^x = (3^x)^2 = y^2 \]
Thus, we can rewrite the inequality as:
\[ -a y^2 + (a - 2)y - \left(\frac{5}{4}a - 1\right) > 0 \]

### Step 2: Rearranging the inequality
Rearranging gives us:
\[ -a y^2 + (a - 2)y - \left(\frac{5}{4}a - 1\right) > 0 \]
This is a quadratic inequality in \(y\).

### Step 3: Identify coefficients
The quadratic can be expressed in the standard form \(Ay^2 + By + C > 0\) where:
- \(A = -a\)
- \(B = a - 2\)
- \(C = -\left(\frac{5}{4}a - 1\right)\)

### Step 4: Conditions for the quadratic to be positive
For the quadratic \(Ay^2 + By + C > 0\) to hold for all \(y\) in the interval \(0 < y < 1\):
1. \(A > 0\) (which means \(-a > 0\) or \(a < 0\), which is satisfied since \(a < 0\))
2. The discriminant \(D = B^2 - 4AC < 0\)

### Step 5: Calculate the discriminant
Now, we calculate the discriminant:
\[ D = (a - 2)^2 - 4(-a)\left(-\left(\frac{5}{4}a - 1\right)\right) \]
This simplifies to:
\[ D = (a - 2)^2 - 4a\left(\frac{5}{4}a - 1\right) \]
Expanding this:
\[ D = (a - 2)^2 - 4a\left(\frac{5}{4}a - 1\right) = (a - 2)^2 - 5a^2 + 4a \]
Now, simplifying further:
\[ D = a^2 - 4a + 4 - 5a^2 + 4a = -4a^2 + 4 \]

### Step 6: Set the discriminant less than zero
We need:
\[ -4a^2 + 4 < 0 \]
This simplifies to:
\[ 4a^2 > 4 \quad \Rightarrow \quad a^2 > 1 \]
Taking square roots gives:
\[ |a| > 1 \]
Since \(a < 0\), we have:
\[ a < -1 \]

### Conclusion
Thus, the value of \(a\) that satisfies the inequality is:
\[ a \in (-\infty, -1) \]