Q.Write the following intervals in set-builder form :

Concept: Interval notation describes a continuous set of real numbers between two endpoints; set-builder form expresses the same set using inequalities and set notation.

Each interval translates by identifying whether endpoints are included (brackets) or excluded (parentheses), then writing the corresponding inequality:

(i) $(−3, 0)$: Both endpoints excluded, so $x$ lies strictly between them.

(ii) $[6, 12]$: Both endpoints included.

(iii) $(6, 12]$: Left endpoint excluded, right included.

$\{x : x \in \mathbb{R}, \, 6 < x \le 12\}$ …

Interval notation is a compact way to describe continuous ranges of real numbers. The symbols tell us which endpoints are included: parentheses $($ and $)$ mean the endpoint is not included (open), while brackets $[$ and $]$ mean it is included (closed). Set-builder notation makes the same statement using inequalities inside set braces, reading as "the set of all $x$ such that…"

The translation follows a simple pattern: look at each endpoint, check whether it's open or closed, and write the corresponding strict ($<$ or $>$) or non-strict ($\leq$ or $\geq$) inequality.

Let me convert each interval:

1. $(−3, 0)$: Both endpoints use parentheses, so both are excluded. The variable $x$ must be strictly greater than $-3$ and strictly less than $0$.

2. $[6, 12]$: Both endpoints use brackets, so both are included. The variable $x$ can equal $6$ or $12$, giving us non-strict inequalities on both sides. $\{x : 6 \leq x \leq 12\} \quad \text{or} \quad \{x \in \mathbb{R} : 6 \leq x \leq 12\}$ …

Concept: Interval Notation

Set-builder form expresses an interval as $\{ x \mid \text{condition on } x \}$, using round brackets ( ) for exclusive endpoints and square brackets [ ] for inclusive endpoints.

Steps

1. Identify endpoints and whether they are included (closed) or excluded (open).
2. Write the condition using inequality signs:
   • ( or ) → < or >
   • [ or ] → ≤ or ≥
3. Combine into $\{ x \mid a < x < b \}$ (or with ≤/≥ as needed).

Answers

(i) $(-3, 0)$ → $\{ x \mid -3 < x < 0 \}$ …

Common Mistakes in Writing Interval Notation

Mistake 1: Confusing Open and Closed Brackets

The Error:

Students often write (–4, 6] as [–4, 6] or (–4, 6), misreading the inequality signs.

Why It Happens:

• < and > mean open (excluded endpoint) → use ( or )
• ≤ and ≥ mean closed (included endpoint) → use [ or ]

How to Avoid:

Draw a number line for each inequality. Mark the endpoint with:

• An open circle (○) for < or >
• A filled circle (●) for ≤ or ≥

Then write the bracket that matches the circle.

Correct answers:

• (i) $(-4, 6]$
• (ii) $(-12, -10)$
• (iii) $[0, 7)$
• (iv) $[3, 4]$

Mistake 2: Reversing the Order of Endpoints

The Error:

Writing (6, –4] instead of (-4, 6] — putting the larger number first.

Why It Happens:

Students copy the order from the set-builder form without thinking about the number line.

How to Avoid:

Always write the smaller number on the left. On a number line, values increase left to right. Interval notation must follow this natural order.

Memory trick:

"Left is less, right is greater — always write small → big."

Mistake 3: Forgetting the Variable is Real Numbers

The Error:

Writing (-4, 6] but thinking it only includes integers like {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Why It Happens:

Students confuse interval notation (continuous set of real numbers) with roster form (listing discrete elements).

How to Avoid:

Remember: $x \in \mathbb{R}$ means all real numbers between the endpoints — including fractions, decimals, and irrationals. The interval (-4, 6] contains infinitely many numbers, not just integers.

Mistake 4: Mixing Up Infinity Notation

The Error:

Writing [–∞, 6] or (–12, ∞] — using closed brackets with infinity.

Why It Happens: …