If $[p]$ means the greatest integer less than or equal to $p$, then $\left[-\frac{1}{4}\right] + \left[4\frac{1}{4}\right] + [3]$ is equal to:

### Concept & Logic This problem introduces the **Greatest Integer Function**, also known as the floor function, denoted by $\lfloor p \rfloor$ or $[p]$. The rule is simple: $[p]$ outputs the largest integer that is less than or equal to $p$. For positive decimals, you effectively drop the decimal. For negative decimals, you must round down to the next most negative integer (moving left on the number line). ### Step-by-Step Solution Evaluate each term individually using the definition of the greatest integer function: * **Step 1: Evaluate $\left[-\frac{1}{4}\right]$** The value $-\frac{1}{4}$ is $-0.25$. The integers surrounding $-0.25$ are $-1$ and $0$. Since the function returns the integer *less* than or equal to the number, we round down to $-1$. $$\left[-\frac{1}{4}\right] = -1$$ * **Step 2: Evaluate $\left[4\frac{1}{4}\right]$** The value $4\frac{1}{4}$ is $4.25$. The greatest integer less than or equal to $4.25$ is $4$. $$\left[4\frac{1}{4}\right] = 4$$ * **Step 3: Evaluate $[3]$** Since $3$ is already an integer, the greatest integer less than or equal to $3$ is simply $3$ itself. $$[3] = 3$$ * **Step 4: Sum the Values** Now, add the three evaluated components together: $$-1 + 4 + 3 = 6$$ ### Exam Strategy & Shortcut Visualize a number line. For any number $p$, $[p]$ simply asks you to "slide left" until you hit the first whole number. For $4.25$, sliding left hits $4$. For $-0.25$, sliding left crosses zero and hits $-1$. ### Common Pitfall The most frequent mistake is assuming that $\left[-\frac{1}{4}\right]$ equals $0$. Students often incorrectly truncate the decimal (dropping the $.25$) instead of finding the integer *less* than the value. Remember, $0$ is greater than $-0.25$, so it violates the rule. ### Final Answer **Therefore, the correct answer is 6.**