Let's solve the inequality \( -|9 - 3x| + 5 > -1 \).

### Step 1: Isolate the absolute value term

First, subtract 5 from both sides to get rid of the constant term on the left side:

\[
- |9 - 3x| > -1 - 5
\]

This simplifies to:

\[
- |9 - 3x| > -6
\]

### Step 2: Divide both sides by -1

To eliminate the negative sign, divide both sides by \(-1\). Remember, dividing by a negative number reverses the inequality:

\[
|9 - 3x| < 6
\]

### Step 3: Solve the absolute value inequality

The inequality \( |9 - 3x| < 6 \) can be split into two inequalities:

\[
-6 < 9 - 3x < 6
\]

### Step 4: Solve for \(x\)

Now solve both parts of the inequality.

#### Left part:
\[
-6 < 9 - 3x
\]
Subtract 9 from both sides:
\[
-15 < -3x
\]
Divide by \(-3\) (and reverse the inequality):
\[
5 > x
\]
or
\[
x < 5
\]

#### Right part:
\[
9 - 3x < 6
\]
Subtract 9 from both sides:
\[
-3x < -3
\]
Divide by \(-3\) (and reverse the inequality):
\[
x > 1
\]

### Step 5: Final solution

The solution to the inequality is:

\[
1 < x < 5
\]

This means \(x\) must be between 1 and 5 (not including 1 and 5).

---

Would you like further details on any step or have additional questions?

Here are 5 questions you might explore next:
1. How do we handle inequalities involving absolute values?
2. What happens when we divide an inequality by a negative number?
3. How do absolute value functions behave graphically?
4. How can inequalities be solved graphically?
5. What are the common mistakes when solving absolute value inequalities?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the inequality −∣9−3x∣+5 > −1 with a detailed, step-by-step solution. This problem involves absolute value inequalities, focusing on isolating the absolute value term and reversing the inequality sign when dividing by a negative number. Suitable for students in Grades 9-11.

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