Linear Inequations MCQ Quiz - Objective Question with Answer for Linear Inequations - Download Free PDF

Given:

The inequation: (x + 2)/(x + 3) > 1

Formula used:

For a rational inequality of the form (a/b) > 1, we analyze critical points and test values between intervals.

Calculation:

(x + 2)/(x + 3) > 1

⇒ (x + 2) - (x + 3) > 0

⇒ x + 2 - x - 3 > 0

⇒ -1 > 0

This is not possible.

Since the inequality is never satisfied, there are no positive integral solutions.

∴ The correct answer is option (4).

Calculation:

Given,

The inequality is: , where k is a natural number.

We need to compute

Approximating √ 2≈ 1.414 , we find:

The inequality becomes:

This implies that k > 12.07 \), so the smallest integer value of k  is:

∴ The value of k  is 13.

Hence, the correct answer is Option 2. 

Calculation: 

x + y = 5λ

Cases :

Total = 120

Hence, the correct answer is 120. 

Calculation: 

(p ∧ ~ q) → (p → ~ q) 

≡ (~ (p ∧ ~q)) ∨ (~ p ∨ ~q)

≡ (~ (p ∨ ~q)) ∨ (~ p ∨ ~q)

≡ ~ p ∨ t ≡ t

∴ The statement (p ∧ (~q)) ⇒ (p ⇒ (~q)) is tautology 

Hence, the correct answer is Option 2. 

Calculation

x² - 4x - 21 > 0:

(x-7)(x+3) > 0.

Solutions: x 7.

x² - 9x + 8

(x-1)(x-8)

Solutions: 1

Combine solutions: x 7, and 1

Intersection is 7

Integral values between 7 and 8: None.

Therefore, no integral values satisfy both inequalities.

Hence option 4 is correct

CALCULATIONS:

Given 

Therefore, option (1) is the correct answer.

Concept:

Comparison using inequality:

For any two real numbers  and  if  then .

Calculation:

Use the given inequality and proceed as follows:

Therefore, we conclude that .

Concept:

Rules for Operations on Inequalities:

• Adding the same number to each side of an inequality does not change the direction of the inequality symbol.
• Subtracting the same number from each side of an inequality does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.
• Dividing each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.

Calculation

Given: 0

Multiply by 2 in above inequality (Here 2 is a positive number so the direction of the inequality does not change)

⇒ 0

⇒ −6 ∴ x lies in (−6, 0)

Concept:

Rules for Operations on Inequalities:

• Adding the same number to each side of an inequality does not change the direction of the inequality symbol.
• Subtracting the same number from each side of an inequality does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.
• Dividing each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.

Calculation:

Given:

Multiplying each side of an inequality by a negative number (-1/3) reverses the direction of the inequality symbol.

, Here x – 1 ≠ 0 ⇒ x ≠ 1

Hence the solution set of the given in equations is (1, 3]

Concept:

Solution Set: The set of all possible values of x.

When (a > 0 and b 0) then 

When (a > 0 and b > 0) or (a ">0\)

Calculation:

Here, 

So, x - 2

⇒ x

∴ x ∈ (-∞ , 2)

Calculation:

Here, constraints: x ≤ 40, y ≤  20 and x, y ≥ 0 

We get minimum value of P, only when x = 0 and y = 0

So, P = 6(0) + 16(0) = 0

Hence, option (3) is correct.

CALCULATION:

Given  \frac{x}{2} + 1\)

1\)

1\)

1\)

1\)

4\)

Therefore option (3) is the correct answer.

Concept:

If f (x) = |x|, then

Calculation:

Given: |x² - 3x + 2| > x² - 3x + 2

Case-1: If x ≥ 0

⇒ x² - 3x + 2 > x² - 3x + 2

∴ No real value of x satisfies the above equation.

Case-2:- If x

⇒ - (x² - 3x + 2) > x² - 3x + 2

⇒ x² - 3x + 2

⇒ (x - 1) (x - 2) ⇒ 1

Concept:

First, graph the "equals" line, then shade in the correct area.

There are three steps:

• Rearrange the equation so "y" is on the left and everything else on the right.
• Plot the "y = " line (make it a solid line for y ≤ or y ≥, and a dashed line for y )
• Shade above the line for a "greater than" (y > or y ≥) or below the line for a "less than" (y

Calculation:

Given: The constraints –x + y ≤ 1, −x + 3y ≤ 9 and x, y ≥ 0

–x + y ≤ 1

⇒ y ≤ 1 + x

Hence shade below the line.

−x + 3y ≤ 9

3y ≤ 9 + x

⇒ y ≤ 3 + x/3

Hence shade below the line.

We can see through above graph region is unbounded feasible space

Concept:

Rules for Operations on Inequalities:

• Adding the same number to each side of an inequality does not change the direction of the inequality symbol.
• Subtracting the same number from each side of an inequality does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.
• Dividing each side of an inequality by a positive number does not change the direction of the inequality symbol.
• Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.

Calculation:

Given:

Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.

0\), Here x ≠ 0

∴ x 12