 open vertical bar, 3, x, plus, 12, close vertical bar, is less than, 9∣3x+12∣<9

The inequality given is \( |3x + 12| < 9 \).

To solve this, follow these steps:

### Step 1: Remove the absolute value
We rewrite the inequality without the absolute value by considering the two cases of the absolute value:

1. \( 3x + 12 < 9 \)
2. \( -(3x + 12) < 9 \) (which is equivalent to \( 3x + 12 > -9 \))

### Step 2: Solve each case separately

#### Case 1: \( 3x + 12 < 9 \)
\[
3x + 12 < 9
\]
Subtract 12 from both sides:
\[
3x < 9 - 12
\]
\[
3x < -3
\]
Now, divide both sides by 3:
\[
x < -1
\]

#### Case 2: \( 3x + 12 > -9 \)
\[
3x + 12 > -9
\]
Subtract 12 from both sides:
\[
3x > -9 - 12
\]
\[
3x > -21
\]
Now, divide both sides by 3:
\[
x > -7
\]

### Step 3: Combine the results
The solution to the inequality is the intersection of the two cases, which gives:
\[
-7 < x < -1
\]

### Final Answer:
The solution is \( -7 < x < -1 \).

Would you like more details on any of the steps or explanations?

Here are 5 related questions for further practice:
1. Solve \( |2x - 5| \leq 7 \).
2. Solve \( |4x + 3| \geq 11 \).
3. Solve \( |x - 2| + 5 > 8 \).
4. Solve \( |5x + 4| \leq 6 \).
5. Solve \( |x + 1| < 10 \).

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

Learn how to solve the absolute value inequality |3x + 12| < 9 with a detailed step-by-step guide. This solution covers key algebraic concepts such as solving inequalities and handling absolute values. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation