https://file.aiquickdraw.com/mathbot/file/1726367894766ixv87i9e.jpg

The inequality you provided in the image is:

\[
(\log_6 x)^2 + 10 < 7 (\log_6 x)
\]

To solve this inequality, we can treat it as a quadratic inequality. Let’s follow these steps:

### Step 1: Substitution
Let:
\[
y = \log_6 x
\]
Now the inequality becomes:
\[
y^2 + 10 < 7y
\]

### Step 2: Rearranging the inequality
Move all terms to one side:
\[
y^2 - 7y + 10 < 0
\]

### Step 3: Solve the quadratic equation
We now solve the quadratic equation:
\[
y^2 - 7y + 10 = 0
\]
Using the quadratic formula:
\[
y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)}
\]
\[
y = \frac{7 \pm \sqrt{49 - 40}}{2}
\]
\[
y = \frac{7 \pm \sqrt{9}}{2}
\]
\[
y = \frac{7 \pm 3}{2}
\]
So the two solutions for \(y\) are:
\[
y_1 = \frac{7 + 3}{2} = 5, \quad y_2 = \frac{7 - 3}{2} = 2
\]

### Step 4: Analyze the quadratic inequality
We have the equation:
\[
y^2 - 7y + 10 < 0
\]
The roots of the corresponding quadratic equation are \(y = 5\) and \(y = 2\). To determine the sign of the expression, we analyze the sign in the intervals defined by these roots: \( (-\infty, 2) \), \( (2, 5) \), and \( (5, \infty) \).

The quadratic expression is negative between the roots, so:
\[
2 < y < 5
\]

### Step 5: Reverse substitution
Since \(y = \log_6 x\), the inequality becomes:
\[
2 < \log_6 x < 5
\]

### Step 6: Solve for \(x\)
To solve for \(x\), rewrite the inequality in exponential form:
\[
6^2 < x < 6^5
\]
\[
36 < x < 7776
\]

### Final Answer:
The solution to the inequality is:
\[
36 < x < 7776
\]

Would you like more details on any of the steps, or do you have any other questions?

### Related Questions:
1. How is the quadratic formula derived?
2. What are the steps for solving exponential inequalities?
3. How do you graph quadratic inequalities?
4. What is the logarithmic function's inverse and how do we apply it to inequalities?
5. How can we solve inequalities involving different logarithmic bases?

### Tip:
When solving logarithmic inequalities, always check if the logarithmic argument is valid (i.e., greater than zero).

Learn how to solve the logarithmic inequality (log_6 x)^2 + 10 < 7(log_6 x) step-by-step. This problem involves logarithmic and quadratic equations and is suitable for high school students in Grades 10-12.

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