For how many integers n is `(7n-26)(5n+11)` a negative number ?

Options

A. None
B. Two
C. Four
D. Six

Answer

To find how many integers \( n \) make the expression \( (7n - 26)(5n + 11) \) negative, we will follow these steps:

### Step 1: Identify the critical points
We need to find when each factor equals zero:
1. Set \( 7n - 26 = 0 \):
\[ 7n = 26 \implies n = \frac{26}{7} \approx 3.71 \]
2. Set \( 5n + 11 = 0 \):
\[ 5n = -11 \implies n = -\frac{11}{5} \approx -2.2 \]

### Step 2: Determine the intervals
The critical points divide the number line into three intervals:
1. \( n < -\frac{11}{5} \)
2. \( -\frac{11}{5} < n < \frac{26}{7} \)
3. \( n > \frac{26}{7} \)

### Step 3: Test each interval
We will test a point from each interval to determine the sign of the product \( (7n - 26)(5n + 11) \).

#### Interval 1: \( n < -\frac{11}{5} \)
Choose \( n = -3 \):
\[ 7(-3) - 26 = -21 - 26 = -47 \quad (\text{negative}) \]
\[ 5(-3) + 11 = -15 + 11 = -4 \quad (\text{negative}) \]
Product: \( (-)(-) = (+) \) (positive)

#### Interval 2: \( -\frac{11}{5} < n < \frac{26}{7} \)
Choose \( n = 0 \):
\[ 7(0) - 26 = -26 \quad (\text{negative}) \]
\[ 5(0) + 11 = 11 \quad (\text{positive}) \]
Product: \( (-)(+) = (-) \) (negative)

#### Interval 3: \( n > \frac{26}{7} \)
Choose \( n = 5 \):
\[ 7(5) - 26 = 35 - 26 = 9 \quad (\text{positive}) \]
\[ 5(5) + 11 = 25 + 11 = 36 \quad (\text{positive}) \]
Product: \( (+)(+) = (+) \) (positive)

### Step 4: Conclusion on the intervals
The product \( (7n - 26)(5n + 11) \) is negative only in the interval \( -\frac{11}{5} < n < \frac{26}{7} \).

### Step 5: Finding the integer solutions
Now we need to find the integers in the interval \( -\frac{11}{5} < n < \frac{26}{7} \):
- \( -\frac{11}{5} \approx -2.2 \) (so the smallest integer is \( -2 \))
- \( \frac{26}{7} \approx 3.71 \) (so the largest integer is \( 3 \))

The integers in this range are:
\[ -2, -1, 0, 1, 2, 3 \]
Counting these gives us a total of \( 6 \) integers.

### Final Answer
Thus, the number of integers \( n \) for which \( (7n - 26)(5n + 11) \) is negative is \( \boxed{6} \).

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