Given that $$ 3 \leq q

Question

Given that $$ 3 \leq q<12 $$ and $$ 4-r<6 $$, where $$ q $$ and $$ r $$ are integers. Find the minimum value of $$ q 	imes r $$.

UpStudy AI · Accepted Answer

**Step 1. Determine the range for $$ q $$:**

We are given  
$$
3 \leq q < 12
$$  
Since $$ q $$ is an integer, the possible values of $$ q $$ are  
$$
q \in \{3,4,5,6,7,8,9,10,11\}.
$$

**Step 2. Solve the inequality for $$ r $$:**

We are given  
$$
4 - r < 6.
$$  
Subtract $$4$$ from both sides:  
$$
-r < 2.
$$  
Multiply both sides by $$-1$$ (remember to reverse the inequality):  
$$
r > -2.
$$  
Since $$ r $$ is an integer, the smallest possible value for $$ r $$ is $$-1$$ (because $$ r $$ must be greater than $$-2$$). Hence,  
$$
r \in \{-1, 0, 1, 2, \dots\}.
$$

**Step 3. Minimize $$ q \times r $$:**

We want the minimum value of the product $$ q \times r $$. Notice that:
- $$ q $$ is positive.
- $$ r $$ can be negative, and multiplying a positive number by a negative number gives a negative product.

To make $$ q \times r $$ as small (i.e., as negative) as possible, we choose:
- The largest possible $$ q $$ (to maximize the magnitude).
- The smallest possible $$ r $$ (the most negative, in this case $$-1$$).

The largest value of $$ q $$ from the set is $$ 11 $$, and the smallest value of $$ r $$ is $$-1$$.

Thus,  
$$
q \times r = 11 \times (-1) = -11.
$$

**Answer:** The minimum value of $$ q \times r $$ is $$-11$$.
The minimum value of $$ q \times r $$ is $$-11$$.

Given that and , where and are
integers. Find the minimum value of .

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