Question: User: . 3\mid10m+4\mid-1\geq41
1. **State the problem:** Solve the inequality $$3|10m+4|-1 \geq 41$$.
2. **Isolate the absolute value:** Add 1 to both sides:
$$3|10m+4| \geq 42$$
3. **Divide both sides by 3:**
$$|10m+4| \geq \cancel{3}\frac{42}{\cancel{3}} = 14$$
4. **Recall the definition of absolute value inequality:**
If $$|x| \geq a$$ for $$a > 0$$, then $$x \leq -a$$ or $$x \geq a$$.
5. **Apply to our inequality:**
$$10m + 4 \leq -14 \quad \text{or} \quad 10m + 4 \geq 14$$
6. **Solve each inequality separately:**
- For $$10m + 4 \leq -14$$:
$$10m \leq -14 - 4$$
$$10m \leq -18$$
$$m \leq \frac{\cancel{10}(-18)}{\cancel{10}} = -1.8$$
- For $$10m + 4 \geq 14$$:
$$10m \geq 14 - 4$$
$$10m \geq 10$$
$$m \geq \frac{\cancel{10}10}{\cancel{10}} = 1$$
7. **Final solution:**
$$m \leq -1.8 \quad \text{or} \quad m \geq 1$$