If `G^2 < G`, which of the following could be G ?

Options

A. 1
B. `23/7`
C. `7/23`
D. `-4`

Answer

To solve the inequality \( G^2 < G \), we can follow these steps:

### Step 1: Rearranging the Inequality
We start with the inequality:
\[ G^2 < G \]
To analyze this inequality, we can rearrange it:
\[ G^2 - G < 0 \]

### Step 2: Factoring the Expression
Next, we factor the left-hand side:
\[ G(G - 1) < 0 \]

### Step 3: Finding the Critical Points
The critical points occur when the expression equals zero:
\[ G(G - 1) = 0 \]
This gives us the solutions:
\[ G = 0 \quad \text{or} \quad G = 1 \]

### Step 4: Testing Intervals
Now we need to test the intervals determined by the critical points \( G = 0 \) and \( G = 1 \). The intervals to test are:
1. \( (-\infty, 0) \)
2. \( (0, 1) \)
3. \( (1, \infty) \)

#### Interval 1: \( (-\infty, 0) \)
Choose a test point, say \( G = -1 \):
\[ (-1)(-1 - 1) = (-1)(-2) = 2 > 0 \quad \text{(not valid)} \]

#### Interval 2: \( (0, 1) \)
Choose a test point, say \( G = 0.5 \):
\[ (0.5)(0.5 - 1) = (0.5)(-0.5) = -0.25 < 0 \quad \text{(valid)} \]

#### Interval 3: \( (1, \infty) \)
Choose a test point, say \( G = 2 \):
\[ (2)(2 - 1) = (2)(1) = 2 > 0 \quad \text{(not valid)} \]

### Step 5: Conclusion
The valid interval where \( G^2 < G \) is:
\[ G \in (0, 1) \]
This means \( G \) must be a number between 0 and 1, excluding 0 and 1.

### Step 6: Identifying Possible Values for \( G \)
Now we can check the given options to see which one falls within the interval \( (0, 1) \):
- Option A: \( 1 \) (not valid, as it is equal to 1)
- Option B: \( \frac{23}{7} \) (approximately 3.29, not valid)
- Option C: \( \frac{7}{23} \) (approximately 0.304, valid)
- Option D: \( -1 \) (not valid, as it is less than 0)

Thus, the only option that satisfies \( G^2 < G \) is:
\[ \text{Option C: } \frac{7}{23} \]