If $x$  is any real number, then the greatest integer which doesn’t exceed $\text{'}x\text{'}$  is called the integral part of $\text{'}x\text{'}$ and will denoted by $\left[x\right]$ .

The integral part of $x$, denoted by $[x]$, is defined as the greatest integer less than or equal to $x$$x$. Below are some examples and properties to help understand and solve questions related to the integral part:

Properties of $[x]$$[$$x$$]$:

1. $[x] \leq x < [x] + 1$
   This means $x$ lies between $[x]$ and $[x] + 1$, where $[x]$ is the largest integer less than or equal to $x$.
2. If $x$ is an integer:
   $[x] = x$.
3. If $x$ is not an integer:
   $[x]$ is the nearest integer less than $x$.
4. For a negative $x$:
   $[x]$ is the closest integer less than $x$.

Examples:

1. If $x = 3.7$$x$$=$$3.7$:
   The greatest integer less than or equal to $3.7$ is $3$.
   $[3.7] = 3$.
2. If $x = -2.3$$x$$=$$-2.3$:
   The greatest integer less than or equal to $-2.3$ is $-3$.
   $[-2.3] = -3$.
3. If $x = 5$$x$$=$$5$:
   $[5] = 5$ (since $5$ is already an integer).
4. If $x = -1.8$$x$$=$$-1.8$:
   $[-1.8] = -2$.

Typical Questions Involving $[x]$:

1. Evaluate $[3.5] + [-2.7]$$[$$3.5$$]$$+$$[$$-2.7$$]$:

• $[3.5] = 3$
• $[-2.7] = -3$
  Thus, $[3.5] + [-2.7] = 3 + (-3) = 0$.

2. Solve $[x] + [2x] = 6$$[$$x$$]$$+$$[$$2$$x$$]$$=$$6$:

• $[x] = n$ (where $n$ is an integer).
• $[2x] = [2n + f]$, where $f$ is the fractional part ($0 \leq f < 1$).

From the condition $[x] + [2x] = 6$:
$n + [2n + f] = 6$.

• Since $[2n + f]$ must also be an integer, analyze $f$ for $n$.
• Solve $6 - n \leq 2n + f < 7 - n$.

3. Prove $[x] + [x + \frac{1}{2}] + [x + \frac{2}{3}] = 3[x]$$[$$x$$]$$+$$[$$x$$+$$21$​$]$$+$$[$$x$$+$$32$​$]$$=$$3$$[$$x$$]$:

This can be approached using properties of $[x]$, analyzing fractional parts, and proving the equality holds.