State whether the following statement is True or False: If $x < -5$ and $x < -2$, then $x \in (-\infty, -5)$.

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

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x < -5

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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 asks for the intersection of two inequalities, $x < -5$ and $x < -2$ . Since $x$ must satisfy both conditions, it must be less than the smaller of the two upper bounds, which is $-5$ . Thus, the solution is $x \in (-\infty, -5)$ , making the statement True.

When we encounter a statement like "If $x < -5$ and $x < -2$ ", we are looking for values of $x$ that satisfy both conditions simultaneously. In mathematics, the word "and" between two conditions implies finding the intersection of their individual solution sets.

Imagine a number line.

The condition $x < -5$ means all numbers to the left of $-5$ , not including $-5$ itself. This can be represented as the interval $(-\infty, -5)$ .
The condition $x < -2$ means all numbers to the left of $-2$ , not including $-2$ itself. This can be represented as the interval $(-\infty, -2)$ .

For $x$ to satisfy both $x < -5$ and $x < -2$ , it must fall into the region where these two intervals overlap. This overlap is precisely the intersection of the two sets.

Let's work through the problem step-by-step.

Identify the individual solution sets:

The first inequality is $x < -5$ .

The solution set for this inequality is all real numbers strictly less than $-5$ . In interval notation, this is $(-\infty, -5)$ .

The second inequality is $x < -2$ .

The solution set for this inequality is all real numbers strictly less than $-2$ . In interval notation, this is $(-\infty, -2)$ .
Understand the logical connector "and":

The statement uses "and", which means we need to find the values of $x$ that satisfy both $x < -5$ AND $x < -2$ . This is equivalent to finding the intersection of the two individual solution sets.

Mathematically, if $S_1$ is the solution set for the first inequality and $S_2$ is the solution set for the second inequality, then the combined solution set is $S_1 \cap S_2$ .

Find the intersection of the solution sets:

We need to find the intersection of $(-\infty, -5)$ and $(-\infty, -2)$ .

Let's visualize this on a number line:

<--------------------------------------------------------------------->
... -6   -5   -4   -3   -2   -1    0    1 ...

x < -5:  <--------------------)
         (--------------------)
         ^
         -5 (not included)

x < -2:  <----------------------------------)
         (----------------------------------)
         ^
         -2 (not included)
``` …

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Q.State whether the following statement is True or False: If x<−5x < -5x<−5 and x<−2x < -2x<−2, then x∈(−∞,−5)x \in (-\infty, -5)x∈(−∞,−5).

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Q.State whether the following statement is True or False: If $x < -5$ and $x < -2$ , then $x \in (-\infty, -5)$ .

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