Example of a compound inequality: 7 ≤ 4x + 3 < 23.

To solve the compound inequality $7 \leq 4x + 3 < 23$, we need to isolate the variable $x$ in the middle. We will perform the same operations on all three parts of the inequality to keep it balanced.

Step 1: Subtract 3 from all three parts.
We want to get rid of the $+3$ next to the $4x$.
$$7 - 3 \leq 4x + 3 - 3 < 23 - 3$$
$$4 \leq 4x < 20$$

Step 2: Divide all three parts by 4.
Now, we need to get $x$ by itself. Since $4x$ means $4$ times $x$, we divide by $4$.
$$\frac{4}{4} \leq \frac{4x}{4} < \frac{20}{4}$$
$$1 \leq x < 5$$

Verification:
Let's check a number inside the range, like $x = 2$.
$4(2) + 3 = 8 + 3 = 11$.
Is $7 \leq 11 < 23$? Yes, that works.

Let's check the lower boundary, $x = 1$.
$4(1) + 3 = 7$.
Is $7 \leq 7 < 23$? Yes, that works (since the symbol is $\leq$).

Let's check the upper boundary, $x = 5$.
$4(5) + 3 = 23$.
Is $7 \leq 23 < 23$? No, because $23$ is not strictly less than $23$. This matches our result that $x$ must be strictly less than $5$.

The solution is all numbers greater than or equal to $1$ and strictly less than $5$.

Final Answer:
1 ≤ x < 5