State whether the following statement is True or False: If $x > -2$ and $x < 9$, then $x \in (-2, 9)$.

> [!TLDR] > The statement is **True**. The condition $x > -2$ and $x -2$ and $x < 9$, we are placing two simultaneous restrictions on $x$: it must be greater than $-2$ **and** less than $9$. This "and" means both conditions hold at once, so $x$ lies strictly between $-2$ and $9$, not including the endpoints. The notation $(-2, 9)$ is just shorthand for exactly that: all real numbers $x$ such that $-2 $ and $ -2$ means $x$ can be any real number greater than $-2$, but not $-2$ itself. On the number line, this is all points to the right of $-2$, with an open circle at $-2$. 2. **Interpret the second condition**: $x < 9$ means $x$ can be any real number less than $9$, but not $9$ itself. On the number line, this is all points to the left of $9$, with an open circle at $9$. 3. **Combine with "and"**: The word "and" means both conditions must be satisfied simultaneously. So $x$ must be greater than $-2$ **and** less than $9$ at the same time. This gives the compound inequality: $$-2 < x < 9$$ 4. **Match to interval notation**: The standard notation for the set $\{x \in \mathbb{R} \mid -2 < x < 9\}$ is $(-2, 9)$. The round brackets indicate that $-2$ and $9$ are **not included** — exactly what the strict inequalities demand. …

State whether the following statement is True or F – Class 11 Mathematics

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x > -2

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Concept understanding — Linear Inequality Solutions

Linear Inequality Solutions – A First Look

Imagine you're standing on a number line. You know exactly where the number 5 is. But what if I asked you to stand on "all numbers greater than 5"? You can't stand on all of them at once — they stretch infinitely to the right. That's the core idea of an inequality: instead of one exact point, you get a whole region of possible values.

A linear inequality is just like a linear equation ( $ax + b = 0$ ), but instead of an equals sign, you have one of these: $<$ , $>$ , $\le$ , or $\ge$ . The solution is not a single number — it's an interval (or a union of intervals) on the number line.

From Equation to Inequality

Start with a simple equation:

2x + 3 = 7

Solve it: $2x = 4 \implies x = 2$ . One point.

Now change it to an inequality:

2x + 3 > 7

Solve it the same way — but the meaning changes. Subtract 3: $2x > 4$ . Divide by 2: $x > 2$ .

The solution is all numbers greater than 2. On a number line, you draw an open circle at 2 (because 2 itself is not included) and shade everything to the right.

Watch out

If you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. For example: $-x < 5$ becomes $x > -5$ . This is the single most common mistake students make.

The Four Types of Solutions

Inequality	Meaning	Number line representation
$x > a$	All numbers strictly greater than $a$	Open circle at $a$ , shade right
$x \ge a$	All numbers greater than or equal to $a$	Closed (filled) circle at $a$ , shade right
$x < a$	All numbers strictly less than $a$	Open circle at $a$ , shade left
$x \le a$	All numbers less than or equal to $a$	Closed circle at $a$ , shade left

The solution set is usually written in interval notation:

$x > 2$ → $(2, \infty)$
$x \le -3$ → $(-\infty, -3]$

Parentheses $($ or $)$ mean the endpoint is not included. Brackets $[$ or $]$ mean it is included.

Solving a Linear Inequality: Step by Step

Solve $3x - 5 \le 7x + 3$ .

Bring variable terms to one side:

$3x - 5 - 7x \le 3$

$-4x - 5 \le 3$
Isolate the variable term:

$-4x \le 8$
Divide by the coefficient (here it's $-4$ , so reverse the sign):

$x \ge -2$

The solution is $x \ge -2$ , or in interval notation: $[-2, \infty)$ .

Tip

Always check your answer by testing a number from the solution set. For $x \ge -2$ , test $x = 0$ : $3(0)-5 = -5$ and $7(0)+3 = 3$ . Is $-5 \le 3$ ? Yes. Now test a number outside, say $x = -3$ : $3(-3)-5 = -14$ and $7(-3)+3 = -18$ . Is $-14 \le -18$ ? No — so the inequality fails, confirming our solution is correct.

Why This Matters …

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The statement is True. The condition $x > -2$ and $x < 9$ is exactly the definition of the open interval $(-2, 9)$ , so the conclusion follows directly.

Concept and Intuition

The heart of this question is understanding the language of intervals — a compact way to describe a range of numbers. When we say $x > -2$ and $x < 9$ , we are placing two simultaneous restrictions on $x$ : it must be greater than $-2$ and less than $9$ . This "and" means both conditions hold at once, so $x$ lies strictly between $-2$ and $9$ , not including the endpoints.

The notation $(-2, 9)$ is just shorthand for exactly that: all real numbers $x$ such that $-2 < x < 9$ . The round parentheses (instead of square brackets) tell us the endpoints are excluded — which matches the strict inequality signs $>$ and $<$ in the original statement.

So the statement is simply saying: "If $x$ is between $-2$ and $9$ (exclusive), then $x$ belongs to the open interval $(-2, 9)$ ." That's a tautology — it's true by definition.

Step-by-Step Reasoning

Interpret the first condition: $x > -2$ means $x$ can be any real number greater than $-2$ , but not $-2$ itself. On the number line, this is all points to the right of $-2$ , with an open circle at $-2$ .
Interpret the second condition: $x < 9$ means $x$ can be any real number less than $9$ , but not $9$ itself. On the number line, this is all points to the left of $9$ , with an open circle at $9$ .
Combine with "and": The word "and" means both conditions must be satisfied simultaneously. So $x$ must be greater than $-2$ and less than $9$ at the same time. This gives the compound inequality:

-2 < x < 9

Match to interval notation: The standard notation for the set $\{x \in \mathbb{R} \mid -2 < x < 9\}$ is $(-2, 9)$ . The round brackets indicate that $-2$ and $9$ are not included — exactly what the strict inequalities demand. …

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Q.State whether the following statement is True or False: If x>−2x > -2x>−2 and x<9x < 9x<9, then x∈(−2,9)x \in (-2, 9)x∈(−2,9).

Concept and Intuition

Step-by-Step Reasoning

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Q.State whether the following statement is True or False: If $x > -2$ and $x < 9$ , then $x \in (-2, 9)$ .

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