https://file.aiquickdraw.com/mathbot/file/1728551079108crdqzopj.jpg

The inequality given in the image is:

\[
|6x - 9| < 11 - 4x
\]

### Step-by-Step Solution:

#### Step 1: Understand the absolute value inequality

We have the absolute value expression \( |6x - 9| \), which means the inequality can be broken into two cases:

1. \( 6x - 9 \geq 0 \) leads to \( 6x - 9 < 11 - 4x \)
2. \( 6x - 9 < 0 \) leads to \( -(6x - 9) < 11 - 4x \)

#### Step 2: Solve each case

##### Case 1: \( 6x - 9 \geq 0 \)

This simplifies to \( x \geq \frac{3}{2} \), i.e., \( x \geq 1.5 \).

Now solve the inequality \( 6x - 9 < 11 - 4x \):

\[
6x - 9 < 11 - 4x
\]

Add \( 4x \) to both sides:

\[
10x - 9 < 11
\]

Add 9 to both sides:

\[
10x < 20
\]

Divide by 10:

\[
x < 2
\]

So for the first case, we have \( \frac{3}{2} \leq x < 2 \), or \( 1.5 \leq x < 2 \). The only integer value in this range is \( x = 2 \).

##### Case 2: \( 6x - 9 < 0 \)

This simplifies to \( x < \frac{3}{2} \), i.e., \( x < 1.5 \).

Now solve the inequality \( -(6x - 9) < 11 - 4x \):

\[
-(6x - 9) < 11 - 4x
\]

This simplifies to:

\[
-6x + 9 < 11 - 4x
\]

Add \( 4x \) to both sides:

\[
-2x + 9 < 11
\]

Subtract 9 from both sides:

\[
-2x < 2
\]

Divide by -2 (and reverse the inequality):

\[
x > -1
\]

So for this case, we have \( -1 < x < \frac{3}{2} \), or \( -1 < x < 1.5 \). The integer values in this range are \( x = 0 \) and \( x = 1 \).

#### Step 3: Combine results

The integer values that satisfy the inequality are \( x = 0 \), \( x = 1 \), and \( x = 2 \).

Thus, the number of integral values of \( x \) that satisfy the inequality is **3**.

---

Would you like more details on any part of this solution?

Here are five related questions to consider:
1. What happens if the inequality sign changes from "<" to "≤" in the equation?
2. How would the solution change if we replaced the absolute value with a different function?
3. Can you solve similar inequalities with different coefficients inside the absolute value term?
4. What if there were two absolute values on both sides of the inequality?
5. How does the method for solving absolute value inequalities differ from solving quadratic inequalities?

**Tip:** When solving absolute value inequalities, always break the problem into two cases (positive and negative) to simplify the solution process.

The number of integral values of x satisfying the inequality |6x - 9| < 11 - 4x is:

This problem demonstrates how to solve an absolute value inequality, |6x - 9| < 11 - 4x, using algebraic techniques. The solution involves breaking the inequality into two cases and solving for integral values. Appropriate for students in Grades 9-12.

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