https://file.aiquickdraw.com/mathbot/file/1726328767920ssk4shq5.jpg

The inequality given is:

\[
|3v + 12| > 3
\]

To solve this inequality, we need to break it down into two cases since it involves absolute value:

### Case 1: \( 3v + 12 > 3 \)
Solve for \(v\):
\[
3v + 12 > 3
\]
Subtract 12 from both sides:
\[
3v > 3 - 12
\]
\[
3v > -9
\]
Now, divide by 3:
\[
v > -3
\]

### Case 2: \( -(3v + 12) > 3 \)
This is equivalent to:
\[
-3v - 12 > 3
\]
Add 12 to both sides:
\[
-3v > 3 + 12
\]
\[
-3v > 15
\]
Now, divide by -3 (remember to flip the inequality sign):
\[
v < -5
\]

### Combined solution:
The solution is:
\[
v > -3 \quad \text{or} \quad v < -5
\]

This means that the inequality holds for \( v \) in the intervals \( (-\infty, -5) \cup (-3, \infty) \).

### Graphing the solution:
- Draw a number line.
- There will be open circles at \( v = -5 \) and \( v = -3 \) since the inequality is strict (does not include equality).
- Shade the region to the left of \( v = -5 \) (for \( v < -5 \)) and the region to the right of \( v = -3 \) (for \( v > -3 \)).

Would you like further details on how to graph this, or do you have any questions?

Here are five related questions you might want to explore:
1. What is the difference between a strict and non-strict inequality in absolute value problems?
2. How do you solve absolute value inequalities with "less than" instead of "greater than"?
3. How do you represent solutions to inequalities on a number line?
4. Can you solve inequalities involving absolute values of quadratic expressions?
5. How does the solution change when the inequality is \( \leq \) instead of \( > \)?

**Tip:** When solving absolute value inequalities, always remember to consider both the positive and negative cases of the expression inside the absolute value symbol!

Graph the solution to the inequality |3v + 12| > 3 on the number line.

Learn how to graph the absolute value inequality |3v + 12| > 3 on a number line. This problem involves breaking down the inequality into two cases and finding the solution set. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation