What is the value of m which satisfies `3m^(2) - 21m + 30 lt 0` ?

Options

A. `m lt 2` or `m gt 5`
B. `m gt 2`
C. `2 lt m lt 5`
D. `m lt 5`

Answer

To solve the inequality \( 3m^2 - 21m + 30 < 0 \), we will follow these steps:

### Step 1: Factor the quadratic expression
We start with the quadratic inequality:
\[ 3m^2 - 21m + 30 < 0 \]

First, we can factor out the common term from the quadratic:
\[ 3(m^2 - 7m + 10) < 0 \]

Next, we need to factor the quadratic \( m^2 - 7m + 10 \). We look for two numbers that multiply to \( 10 \) and add to \( -7 \). The numbers \( -5 \) and \( -2 \) fit this requirement:
\[ m^2 - 7m + 10 = (m - 5)(m - 2) \]

So we rewrite the inequality:
\[ 3(m - 5)(m - 2) < 0 \]

### Step 2: Analyze the sign of the product
Now we need to determine when the product \( (m - 5)(m - 2) < 0 \). This means we need to find the intervals where the product is negative.

### Step 3: Find the critical points
The critical points are found by setting each factor to zero:
1. \( m - 5 = 0 \) → \( m = 5 \)
2. \( m - 2 = 0 \) → \( m = 2 \)

These points divide the number line into three intervals:
1. \( (-\infty, 2) \)
2. \( (2, 5) \)
3. \( (5, \infty) \)

### Step 4: Test each interval
We will test a point from each interval to see where the product is negative.

1. **Interval \( (-\infty, 2) \)**: Choose \( m = 0 \)
\[ (0 - 5)(0 - 2) = (-5)(-2) = 10 > 0 \]

2. **Interval \( (2, 5) \)**: Choose \( m = 3 \)
\[ (3 - 5)(3 - 2) = (-2)(1) = -2 < 0 \]

3. **Interval \( (5, \infty) \)**: Choose \( m = 6 \)
\[ (6 - 5)(6 - 2) = (1)(4) = 4 > 0 \]

### Step 5: Conclusion
The product \( (m - 5)(m - 2) < 0 \) is satisfied in the interval \( (2, 5) \). Therefore, the solution to the inequality \( 3m^2 - 21m + 30 < 0 \) is:
\[ 2 < m < 5 \]

### Final Answer
The values of \( m \) that satisfy the inequality are \( m \in (2, 5) \).

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