1. The problem is to solve the equation $|2x + 5| = 19$ for $x$. 2. Recall that the absolute value equation $|A| = B$ implies two cases: $A = B$ or $A = -B$, provided $B \geq 0$. 3. Here, $A = 2x + 5$ and $B = 19$, which is positive, so we proceed with both cases. 4. Case 1: $2x + 5 = 19$ Subtract 5 from both sides: $$2x + 5 - 5 = 19 - 5$$ $$2x = 14$$ Divide both sides by 2: $$\frac{\cancel{2}x}{\cancel{2}} = \frac{14}{2}$$ $$x = 7$$ 5. Case 2: $2x + 5 = -19$ Subtract 5 from both sides: $$2x + 5 - 5 = -19 - 5$$ $$2x = -24$$ Divide both sides by 2: $$\frac{\cancel{2}x}{\cancel{2}} = \frac{-24}{2}$$ $$x = -12$$ 6. Therefore, the solutions to the equation $|2x + 5| = 19$ are $x = 7$ and $x = -12$. Final answer: $x = 7$ or $x = -12$.