Given Pxq = 1/2 P+q. Find: a) 6x4, b) -4x6, c) 3x1, d) -3/4 x 2/3

Here is the solution to the problem shown in the image.

: Given the binary operation: $P \times Q = \frac{1}{2} P + Q$. We need to evaluate this operation for the following expressions:

a) Evaluate $6 \times 4$

Step 1: Identify the values for $P$ and $Q$. For $6 \times 4$, we have $P = 6$ and $Q = 4$.

Step 2: Substitute these values into the operation definition. $6 \times 4 = \frac{1}{2}(6) + 4$

Step 3: Perform the multiplication. $6 \times 4 = 3 + 4$

Step 4: Perform the addition. $6 \times 4 = \boxed{7}$

b) Evaluate $-4 \times 6$

Step 1: Identify the values for $P$ and $Q$. For $-4 \times 6$, we have $P = -4$ and $Q = 6$.

Step 2: Substitute these values into the operation definition. $-4 \times 6 = \frac{1}{2}(-4) + 6$

Step 3: Perform the multiplication. $-4 \times 6 = -2 + 6$

Step 4: Perform the addition. $-4 \times 6 = \boxed{4}$

c) Evaluate $3 \times 1$

Step 1: Identify the values for $P$ and $Q$. For $3 \times 1$, we have $P = 3$ and $Q = 1$.

Step 2: Substitute these values into the operation definition. $3 \times 1 = \frac{1}{2}(3) + 1$

Step 3: Perform the multiplication. $3 \times 1 = \frac{3}{2} + 1$

Step 4: Convert to a common denominator and add. $3 \times 1 = \frac{3}{2} + \frac{2}{2}$ $3 \times 1 = \frac{3 + 2}{2}$ $3 \times 1 = \boxed{\frac{5}{2}}$

d) Evaluate $-\frac{3}{4} \times \frac{2}{3}$

Step 1: Identify the values for $P$ and $Q$. For $-\frac{3}{4} \times \frac{2}{3}$, we have $P = -\frac{3}{4}$ and $Q = \frac{2}{3}$.

Step 2: Substitute these values into the operation definition. $-\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}\left(-\frac{3}{4}\right) + \frac{2}{3}$

Step 3: Perform the multiplication. $-\frac{3}{4} \times \frac{2}{3} = -\frac{3}{8} + \frac{2}{3}$

Step 4: Find a common denominator (24) and add the fractions. $-\frac{3}{4} \times \frac{2}{3} = -\frac{3 \times 3}{8 \times 3} + \frac{2 \times 8}{3 \times 8}$ $-\frac{3}{4} \times \frac{2}{3} = -\frac{9}{24} + \frac{16}{24}$ $-\frac{3}{4} \times \frac{2}{3} = \frac{16 - 9}{24}$ $-\frac{3}{4} \times \frac{2}{3} = \boxed{\frac{7}{24}}$