Animated Solution for Mathematics - Basic Mathematics: Let $A=\{x\in \mathbb{R}:|x+1|<2\}$ and $B=\{x\in \mathbb{R}:|x-1|\ge 2\}$. Then which one of the following statements is NOT true?

Imagine you are standing on a vast, infinite number line. This is the stage for our problem today. We are dealing with two sets, $A$ and $B$, defined by absolute value inequalities.

These aren't just abstract symbols; they are geometric descriptions of distance. Let us embark on this journey to decode them.

Decoding Set $A$

We are given $A=\{x\in \mathbb{R}:|x+1|<2\}$. In the language of physics and geometry, $|x+1|$ is the distance between the point $x$ and the point $-1$.

The inequality $|x+1|<2$ tells us that this distance must be strictly less than $2$. If you stand at $-1$ and walk $2$ units to the right, you reach $1$. If you walk $2$ units to the left, you reach $-3$.

Thus, any point $x$ that satisfies this condition must lie strictly between $-3$ and $1$. Mathematically, we write this as:

Subtracting $1$ from all parts, we get $-3<x<1$. So, our Set $A$ is the open interval $(-3,1)$. Visualize this as a blue segment on our number line, with open circles at $-3$ and $1$.

The Split Personality of Set $B$

Now, consider $B=\{x\in \mathbb{R}:|x-1|\ge 2\}$. Here, the distance between $x$ and $1$ is at least $2$.

This is a 'greater than' inequality, which creates a split. Either the distance is to the right of $1$ (i.e., $x-1\ge 2$) or to the left of $1$ (i.e., $x-1\le -2$).

Solving these, we get $x\ge 3$ or $x\le -1$. In interval notation, Set $B$ is $(-\infty ,-1]\cup [3,\infty )$. This is a green segment on our number line, with solid dots at $-1$ and $3$, extending outwards to infinity.

The Art of Set Operations

Now that we have our sets, we can evaluate the options. Let us look at the intersection $A\cap B$.

This is the region where our blue segment $(-3,1)$ and green segment $(-\infty ,-1]\cup [3,\infty )$ overlap. The overlap is clearly the interval $(-3,-1]$. This confirms that statement (C) is true.

Next, consider $A-B$. This means we take the blue segment $(-3,1)$ and remove any part that is covered by the green segment.

The part of $A$ that overlaps with $B$ is $(-3,-1]$. Removing this leaves us with $(-1,1)$. This confirms that statement (A) is true.

Now, let us look at $B-A$. We take the green segments and remove any part that overlaps with the blue segment $(-3,1)$.

The overlap is $(-3,-1]$. Removing this from $B$ leaves $(-\infty ,-3]\cup [3,\infty )$. This is equivalent to $\mathbb{R}-(-3,3)$.

However, statement (B) claims $B-A=\mathbb{R}-(-3,1)$. This is a mismatch! The interval being excluded is wrong. Thus, statement (B) is the one that is NOT true.

Finally, for $A\cup B$, we combine both sets. The union covers everything from $-\infty$ up to $1$, and then from $3$ to $\infty$.

The only gap is the interval $[1,3)$. So, $A\cup B=\mathbb{R}-[1,3)$. This confirms statement (D) is true.

By visualizing these sets as segments on a number line, we transformed a daunting algebraic problem into a simple exercise of overlapping intervals. We found that statement (B) is the false one.

Remember, in JEE, visualization is your greatest weapon. Keep practicing, and the math will start to feel like a story you are writing yourself!