https://r2.erweima.ai/mathbot/image/1726247028918kn2s99n0.jpg

The inequality shown in the image is:

\[
\frac{1}{x + 2} > \frac{2}{x - 1}
\]

To solve this inequality and provide the number line representation, let's follow these steps:

### Step 1: Cross-Multiply
First, cross-multiply to eliminate the fractions, but remember that we need to consider the sign changes depending on the values of \( x \).

\[
(x - 1) \cdot 1 > (x + 2) \cdot 2
\]

Simplify the equation:

\[
x - 1 > 2x + 4
\]

### Step 2: Simplify the Inequality
Rearrange the terms to isolate \( x \):

\[
x - 2x > 4 + 1
\]

\[
-x > 5
\]

Multiply by \(-1\) (which reverses the inequality sign):

\[
x < -5
\]

### Step 3: Consider the Domain Restrictions
Since the inequality involves fractions, \( x \) must not make the denominators zero:

1. \( x + 2 \neq 0 \implies x \neq -2 \)
2. \( x - 1 \neq 0 \implies x \neq 1 \)

### Step 4: Consider Critical Points and Test Intervals
The critical points from the inequality and domain restrictions are \( -5 \), \( -2 \), and \( 1 \).

Test intervals:

1. \( x < -5 \)
2. \( -5 < x < -2 \)
3. \( -2 < x < 1 \)
4. \( x > 1 \)

Evaluate the inequality \( \frac{1}{x + 2} > \frac{2}{x - 1} \) in each interval:

- For \( x < -5 \): True.
- For \( -5 < x < -2 \): False.
- For \( -2 < x < 1 \): Evaluate carefully; generally False.
- For \( x > 1 \): False.

### Step 5: Graph the Solution on a Number Line
The solution is \( x < -5 \), and it does not include \( -2 \) or \( 1 \). 

### Number Line Representation:
```
<====(-∞)----(-5)----(-2)----(1)====>
        O
```

- \( O \) indicates an open interval not including \( -5 \).
- The shaded area is left of \(-5\) (going towards \( -\infty \)).

Would you like further details or have any specific questions?

### Expanding Questions:
1. How do you handle inequalities involving fractions in general?
2. Why do we need to be careful with domain restrictions in inequalities?
3. What happens when multiplying or dividing both sides of an inequality by a negative number?
4. Can you describe how to graph the solution of an inequality on a number line?
5. How would the solution change if the inequality was \( \geq \) instead of \( > \)?

**Tip:** Always remember to check for critical points and domain restrictions when dealing with inequalities involving fractions!

1/(x + 2) > 2/(x - 1) dan berikan garis bilangan

Understand how to solve the inequality 1/(x + 2) > 2/(x - 1) and graph its solution on a number line. This explanation covers the cross-multiplication method, domain restrictions, and properties of inequalities. Suitable for Grades 9-10 students.

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