Solve the inequality
\[2 - \frac{1}{2x + 3} < 4.\]
The given inequality is equivalent to
\[\frac{1}{2x + 3} + 2 > 0,\]or
\[\frac{4x + 7}{2x + 3} > 0.\]If $x < -\frac{7}{4},$ then $4x + 7 < 0$ and $2x + 3 < 0,$ so the inequality is satisfied.

If $-\frac{7}{4} < x < -\frac{3}{2},$ then $4x + 7 > 0$ and $2x + 3 < 0,$ so the inequality is not satisfied.

If $x > -\frac{3}{2},$ then $4x + 7 > 0$ and $2x + 3 > 0,$ so the inequality is satisfied.  Thus, the solution is
\[x \in \boxed{\left( -\infty, -\frac{7}{4} \right) \cup \left( -\frac{3}{2}, \infty \right)}.\]
\le(-\iny,-\frac{7}{4}\right)\cup\le(-\frac{3}{2},\iny\right)
Level 4
Intermediate Algebra
Problem: Solve the inequality
\[2 - \frac{1}{2x + 3} < 4.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the inequality, first isolate the fraction. Start by subtracting 2 from both sides:

-1/2x + 3 < 2

This simplifies to:

-1/2x < 2 - 2
-1/2x < 0

Next, multiply both sides by -1, remembering to reverse the inequality:

1/2x > 0

This implies that x must be positive. 

Now, return to the original inequality and rearrange it:

2 - 4 < 1/2x + 3
-2 < 1/2x + 3

Subtract 3 from both sides:

-2 - 3 < 1/2x
-5 < 1/2x

Multiply both sides by 2:

-10 < x

This gives us the condition that x > -10. 

Next, consider the term -1/(2x + 3). This term is undefined when 2x + 3 = 0, which occurs at x = -3/2. Therefore, x cannot equal -3/2.

Now, combine the conditions derived from the inequality and the undefined point. The solution must satisfy both x > -10 and x cannot equal -3/2. 

The intervals can be expressed as:

1. From -10 to -3/2, which is (-10, -3/2).
2. From -3/2 to infinity, which is (-3/2, infinity).

Thus, the final answer is the union of these intervals:

(-10, -3/2) ∪ (-3/2, infinity).<|END_OF_LATENT|> The final answer is $\le(-\iny,-\frac{7}{4}\right)\cup\le(-\frac{3}{2},\iny\right)$.