To solve the inequality \( \log_{(2x + 3)}(x^2) < \log_{(2x + 3)}(2x + 3) \), we will follow these steps:
### Step 1: Determine the domain of the logarithm
The logarithmic function is defined only for positive arguments. Therefore, we need to ensure that \( 2x + 3 > 0 \).
**Hint:** Solve the inequality \( 2x + 3 > 0 \) to find the range of \( x \).
\[
2x + 3 > 0 \implies 2x > -3 \implies x > -\frac{3}{2}
\]
### Step 2: Analyze the inequality
We can rewrite the inequality as:
\[
\log_{(2x + 3)}(x^2) < \log_{(2x + 3)}(2x + 3)
\]
Since the base \( 2x + 3 > 1 \) (which is true for \( x > -1 \)), we can drop the logarithms and compare the arguments:
\[
x^2 < 2x + 3
\]
**Hint:** Rearrange the inequality to form a standard quadratic inequality.
### Step 3: Rearrange to form a quadratic inequality
Rearranging gives:
\[
x^2 - 2x - 3 < 0
\]
**Hint:** Factor the quadratic expression.
### Step 4: Factor the quadratic
Factoring the quadratic:
\[
(x - 3)(x + 1) < 0
\]
**Hint:** Use the sign chart method to find the intervals where the product is negative.
### Step 5: Determine the intervals
The critical points are \( x = -1 \) and \( x = 3 \). We test intervals around these points:
- For \( x < -1 \) (e.g., \( x = -2 \)): \( (-2 - 3)(-2 + 1) = (-5)(-1) > 0 \)
- For \( -1 < x < 3 \) (e.g., \( x = 0 \)): \( (0 - 3)(0 + 1) = (-3)(1) < 0 \)
- For \( x > 3 \) (e.g., \( x = 4 \)): \( (4 - 3)(4 + 1) = (1)(5) > 0 \)
Thus, the solution to the inequality \( (x - 3)(x + 1) < 0 \) is:
\[
-1 < x < 3
\]
### Step 6: Combine with the domain
From Step 1, we have \( x > -\frac{3}{2} \). Therefore, the combined solution is:
\[
-\frac{3}{2} < x < 3
\]
**Hint:** Identify the intervals and their endpoints.
### Final Step: Identify the values of A, B, C, D
From the solution, we can identify:
- \( A = -\frac{3}{2} \)
- \( B = -1 \)
- \( C = 0 \)
- \( D = 3 \)
The final answer can be expressed as:
\[
(a, b) \cup (b, c) \cup (c, d)
\]
### Summary of Results
The values of \( A, B, C, D \) are:
- \( A = -\frac{3}{2} \)
- \( B = -1 \)
- \( C = 0 \)
- \( D = 3 \)
