Solve the inequality -2|3m| + 3 < -51.

Step-by-step solution

1. 1

   Isolate the absolute value before touching the bars. Subtract $3$ from both sides:

   $-2|3m|<-54$

   The bars have to stand alone (with coefficient $1$) before the case split is legal.

2. 2

   Divide by $-2$ and flip the inequality sign. Dividing an inequality by a negative number reverses its direction:

   $|3m|>27$

   Forgetting this flip is the single most common mistake in this problem — it would turn an outward inequality into an inward one and produce the opposite answer.

3. 3

   Split a 'greater than' absolute value outward. For $b>0$, $|A|>b$ means $A$ is farther than $b$ from zero in either direction:

   $3m<-27\qquad\text{or}\qquad 3m>27$

   (Contrast with $|A|<b$, which gives the single sandwich $-b<A<b$.)

4. 4

   Divide each branch by $3$. Since $3>0$ no further flipping occurs:

   $m<-9\qquad\text{or}\qquad m>9$

5. 5

   Test one value per region. At $m=10$: $-2|30|+3=-57<-51$ $\checkmark$. At $m=0$: $-2\cdot 0+3=3\not<-51$ $\checkmark$ (correctly excluded). So the solution set is $(-\infty,-9)\cup(9,\infty)$.