Solve the inequalities for real x and represent solution on number line
`-5le(2-3x)/(4)le9`,`xinR`

Answer

To solve the inequality \(-5 \leq \frac{2 - 3x}{4} \leq 9\) and represent the solution on a number line, we will follow these steps:

### Step 1: Break down the compound inequality
We can separate the compound inequality into two parts:
1. \(-5 \leq \frac{2 - 3x}{4}\)
2. \(\frac{2 - 3x}{4} \leq 9\)

### Step 2: Solve the first inequality
Start with the first inequality:
\[ -5 \leq \frac{2 - 3x}{4} \]
Multiply all sides by 4 (since 4 is positive, the inequality sign remains the same):
\[ -20 \leq 2 - 3x \]
Now, subtract 2 from both sides:
\[ -20 - 2 \leq -3x \implies -22 \leq -3x \]
Next, divide by -3 (remember to reverse the inequality sign):
\[ \frac{-22}{-3} \geq x \implies x \leq \frac{22}{3} \]

### Step 3: Solve the second inequality
Now solve the second part:
\[ \frac{2 - 3x}{4} \leq 9 \]
Again, multiply all sides by 4:
\[ 2 - 3x \leq 36 \]
Subtract 2 from both sides:
\[ -3x \leq 34 \]
Now divide by -3 (reverse the inequality sign):
\[ x \geq \frac{-34}{-3} \implies x \geq \frac{34}{3} \]

### Step 4: Combine the results
From the two inequalities, we have:
\[ \frac{-34}{3} \leq x \leq \frac{22}{3} \]

### Step 5: Represent the solution on a number line
Now we need to represent the solution \(\frac{-34}{3} \leq x \leq \frac{22}{3}\) on a number line.

1. Calculate the decimal values:
- \(\frac{-34}{3} \approx -11.33\)
- \(\frac{22}{3} \approx 7.33\)

2. Draw a number line and mark the points \(-11.33\) and \(7.33\). Since the inequality includes equal signs, we will use closed circles at both points to indicate that these values are included in the solution.

3. Shade the region between \(-11.33\) and \(7.33\) to indicate all the values of \(x\) that satisfy the inequality.

### Final Solution
The solution set is:
\[ x \in \left[\frac{-34}{3}, \frac{22}{3}\right] \]
And on the number line, it is represented as a shaded region between \(-11.33\) and \(7.33\) with closed circles at both ends.

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