Solve `log_(1//3)(x^(2)-3x+5)lt -1`

Answer

To solve the inequality \( \log_{1/3}(x^2 - 3x + 5) < -1 \), we will follow these steps:

### Step 1: Rewrite the logarithmic inequality
We know that \( \log_{1/3}(a) < -1 \) can be rewritten using the properties of logarithms. Since the base \( \frac{1}{3} \) is less than 1, the inequality will flip when we exponentiate both sides. Thus, we can rewrite the inequality as:
\[ x^2 - 3x + 5 > 3^{-1} \]
Calculating \( 3^{-1} \) gives us \( \frac{1}{3} \). Therefore, the inequality becomes:
\[ x^2 - 3x + 5 > \frac{1}{3} \]

### Step 2: Clear the fraction
To eliminate the fraction, we can multiply through by 3 (which is positive, so the inequality sign remains the same):
\[ 3(x^2 - 3x + 5) > 1 \]
This simplifies to:
\[ 3x^2 - 9x + 15 > 1 \]

### Step 3: Rearranging the inequality
Now, we can rearrange the inequality:
\[ 3x^2 - 9x + 14 > 0 \]

### Step 4: Factor the quadratic expression
Next, we need to factor the quadratic expression \( 3x^2 - 9x + 14 \). To do this, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3, b = -9, c = 14 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-9)^2 - 4(3)(14) = 81 - 168 = -87 \]
Since the discriminant is negative, the quadratic has no real roots.

### Step 5: Determine the sign of the quadratic
Since the leading coefficient (3) is positive and there are no real roots, the quadratic \( 3x^2 - 9x + 14 \) is always positive for all real \( x \).

### Conclusion
Thus, the solution to the inequality \( \log_{1/3}(x^2 - 3x + 5) < -1 \) is:
\[ \text{All real numbers } x \in (-\infty, \infty) \]