Q.Write the following as intervals :

Concept: Interval Notation — a compact way to write a set of real numbers between two endpoints, using ( or ) for strict inequalities and [ or ] for inclusive ones.

Step 1: For (i), $-4 < x \leq 6$ means $x$ is greater than $-4$ (strict) and up to $6$ (inclusive).

Step 2: For (ii), $-12 < x < -10$ is strict at both ends. …

The Concept: Why Interval Notation Works

When you see a set written like $\{x : x \in \mathbb{R}, -4 < x \leq 6\}$, it describes every real number between $-4$ and $6$, but with a specific rule at each end. The inequality $-4 < x$ means $-4$ itself is not part of the set — the set starts just to the right of $-4$. The inequality $x \leq 6$ means $6$ is included.

Interval notation was invented to capture this information in one glance. A round bracket ( or ) means "this endpoint is not included" (open interval). A square bracket [ or ] means "this endpoint is included" (closed interval). The smaller number always goes on the left.

Step-by-Step Solutions

1. (i) $\{x : x \in \mathbb{R}, -4 < x \leq 6\}$

Look at the left end: $-4 < x$. The number $-4$ is not in the set. So we use a round bracket: $(-4$.

Look at the right end: $x \leq 6$. The number $6$ is in the set. So we use a square bracket: $6]$.

Put them together: $(-4, 6]$.

2. (ii) $\{x : x \in \mathbb{R}, -12 < x < -10\}$

Left end: $-12 < x$ means $-12$ is not included → round bracket: $(-12$.

Right end: $x < -10$ means $-10$ is not included → round bracket: $-10)$.

Together: $(-12, -10)$. …

Method: Converting Set-Builder Inequalities to Interval Notation

Method Name: Endpoint-and-Bracket Method

Why This Works

A set like $\{x : x \in \mathbb{R},\ a < x \le b\}$ describes every real number strictly between two endpoints, with each endpoint either included or excluded. Interval notation records the two endpoints and, via the bracket style, whether each one is included.

Steps

Step 1: Identify the two endpoints

Read off the smaller number $a$ and the larger number $b$ from the inequality.

Step 2: Check each inequality symbol

• A strict inequality ($<$ or $>$) means that endpoint is excluded — use a round bracket ( or ).
• A non-strict inequality ($\le$ or $\ge$) means that endpoint is included — use a square bracket [ or ].

Step 3: Write the interval …

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: …