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To solve the nonlinear inequality \( \frac{x - 3}{2x + 1} < 4 \), we first eliminate the fraction by multiplying both sides of the inequality by \(2x + 1\) and being mindful of the sign of the expression:

1. Multiply both sides by \(2x + 1\), remembering that the inequality changes direction if \(2x + 1\) is negative:
   \[
   (x - 3) < 4(2x + 1)
   \]

2. Distribute and simplify:
   \[
   x - 3 < 8x + 4
   \]
   \[
   -3 - 4 < 8x - x
   \]
   \[
   -7 < 7x
   \]
   \[
   x > -1
   \]

3. We must also consider the point where the denominator \(2x + 1\) is zero, as this will be undefined in the original inequality:
   \[
   2x + 1 = 0 \implies x = -\frac{1}{2}
   \]

4. The critical points to consider are \(x = -\frac{1}{2}\). We need to test values around this point to determine where the inequality holds:

   - When \(x < -\frac{1}{2}\), \(2x + 1\) is negative, so we need to reverse the inequality if we multiply by it.
   - When \(x > -\frac{1}{2}\), \(2x + 1\) is positive, so the inequality retains its direction.

5. By solving the inequality \(x > -1\), we note that the original inequality is undefined at \(x = -\frac{1}{2}\), hence this value is excluded from the interval. However, \(x > -1\) includes values from \(-1\) to \(\infty\) except \(-\frac{1}{2}\).

Therefore, the solution to the inequality in interval notation, excluding the point where the denominator is zero, is:
   \[
   \boxed{(-1, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)}
   \]

This interval represents all \(x\) such that \(\frac{x-3}{2x+1} < 4\), except for \(x = -\frac{1}{2}\) where the expression is undefined.

Would you like more details on any part of this solution, or do you have any questions about similar problems?

1. Would you like to see a graphical interpretation of this inequality?
2. Do you need to know how to check your solution for this inequality?
3. Are there specific types of inequalities you need help with, such as polynomial or rational inequalities?
4. Interested in learning about inequalities involving absolute values?
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**Tip:** When solving inequalities involving fractions, always consider the restrictions on the domain caused by the denominator, since dividing or multiplying by a negative number flips the inequality.

Solve the nonlinear inequality. Express the solution using interval notation. (x - 3)/(2x + 1) < 4

Learn how to solve the nonlinear inequality (x - 3)/(2x + 1) < 4 step by step, including consideration of undefined points and sign changes. This detailed solution covers algebraic concepts and is suitable for high school students in Grades 9-12.

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