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Can you answer this step by step, 1 to 4? Step No. 1, rewrite the inequality as a single rational expression. Step 2, determine the meaningful numbers and draw a number line. Step 3, construct the table of values. Step 4, write the solution and Mark it on the number line.

Let's work through the inequality step by step as requested.

### Given Inequality:
\[
\frac{2x - 5}{x - 5} \leq 3
\]

### Step 1: Rewrite the inequality as a single rational expression.

First, subtract 3 from both sides to have everything on one side of the inequality:
\[
\frac{2x - 5}{x - 5} - 3 \leq 0
\]

Now, express the 3 as a fraction with the same denominator:
\[
\frac{2x - 5}{x - 5} - \frac{3(x - 5)}{x - 5} \leq 0
\]

Combine the fractions:
\[
\frac{2x - 5 - 3(x - 5)}{x - 5} \leq 0
\]

Simplify the numerator:
\[
2x - 5 - 3x + 15 = -x + 10
\]

So, the inequality becomes:
\[
\frac{-x + 10}{x - 5} \leq 0
\]
Or equivalently:
\[
\frac{10 - x}{x - 5} \leq 0
\]

### Step 2: Determine the meaningful numbers and draw a number line.

The rational expression \(\frac{10 - x}{x - 5}\) can change signs at the points where the numerator or the denominator equals zero. Therefore, we need to determine the critical values by setting the numerator and denominator equal to zero:

1. **Numerator:** \(10 - x = 0\) → \(x = 10\)
2. **Denominator:** \(x - 5 = 0\) → \(x = 5\)

These values divide the number line into intervals.

### Step 3: Construct the table of values.

We'll check the sign of the expression \(\frac{10 - x}{x - 5}\) in each interval:
1. \(x < 5\)
2. \(5 < x < 10\)
3. \(x > 10\)

| Interval   | Test Point | Sign of \(10 - x\) | Sign of \(x - 5\) | Sign of \(\frac{10 - x}{x - 5}\) |
|------------|------------|--------------------|-------------------|---------------------------------|
| \(x < 5\)  | \(x = 0\)   | Positive           | Negative          | Negative                        |
| \(5 < x < 10\) | \(x = 7\) | Positive           | Positive          | Positive                        |
| \(x > 10\) | \(x = 15\)  | Negative           | Positive          | Negative                        |

### Step 4: Write the solution and mark it on the number line.

Since we want \(\frac{10 - x}{x - 5} \leq 0\), we need to consider where the expression is negative or zero. From the table, the expression is negative in the intervals \(x < 5\) and \(x > 10\).

At \(x = 10\), the expression equals 0, which is included because the inequality is non-strict (\(\leq\)).

Therefore, the solution is:
\[
(-\infty, 5) \cup [10, \infty)
\]

Mark this solution on the number line:

- Open circle at \(x = 5\) (because \(x = 5\) is not included).
- Closed circle at \(x = 10\) (because \(x = 10\) is included).
- The intervals \(x < 5\) and \(x > 10\) are shaded.

---

Would you like more details on any step or have any questions? Here are some related questions to deepen your understanding:

1. How do you handle inequalities when the denominator could be zero?
2. What happens if the inequality had been \(\geq\) instead of \(\leq\)?
3. How do you check if a solution satisfies the original inequality?
4. Why is the sign of the intervals important when solving rational inequalities?
5. What changes if you have more complex numerators or denominators?

**Tip:** Always check if the solution includes points where the expression might be undefined.

Explore the step-by-step solution for the rational inequality \(\frac{2x - 5}{x - 5} \leq 3\). Learn how to rewrite the inequality, draw a number line, construct a table of values, and mark the solution. Suitable for high school students studying inequalities and rational expressions.

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