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

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

Concept:

Solution Region of an Inequality:

• The solution region of an inequality is the set of all points that satisfy the inequality.
• For linear inequalities, the solution region is typically a half-plane or a region bounded by lines.
• The inequality given is: 2x + 4y ≤ 9.
• We can rewrite the inequality as a line equation: 2x + 4y = 9 and plot it on the coordinate plane.
• The solution region consists of all points that satisfy the inequality, which is typically one side of the line.

Calculation:

Given the inequality: 2x + 4y ≤ 9

First, rewrite the inequality as the equation of a line:

2x + 4y = 9

Now, solve for y:

4y = 9 - 2x

y = (9 - 2x) / 4

y = 9/4 - x/2

The slope of the line is -1/2 and the y-intercept is 9/4.

Plot the line y = (9 - 2x)/4 on the coordinate plane.

The solution region will be the area below this line since the inequality is ≤ (i.e., points that satisfy the inequality lie below or on the line).

Thus, the solution region is the half-plane below the line 2x + 4y = 9, including the line itself.

∴ The solution region is the region below and on the line 2x + 4y = 9.

Calculation

Given;

Inequality: 37 - (3x + 5) ≥ 9x - 8(x - 3)

⇒ 37 - 3x - 5 ≥ 9x - 8x + 24

⇒ 32 - 3x ≥ x + 24

⇒ 32 - 24 ≥ x + 3x

⇒ 8 ≥ 4x

⇒ 4x ≤ 8

⇒ x ≤ 2

∴ The solution set is (-∞, 2].

Hence option 3 is correct.

Calculation

Given:

⇒ 

⇒ 

⇒  x \geq \frac{11}{-3}\)

⇒   x \geq -\frac{11}{3}\)

⇒ 

∴ x lies in the interval

Hence option 1 is correct

Concept Used:

The absolute value inequality  can be written as .

Calculation

Given: The inequality is

⇒

⇒

⇒

⇒ 

⇒

∴ The solution of the inequality is .

Hence, option 1 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:

Given  \frac{x}{2} + 1\)

1\)

1\)

1\)

1\)

4\)

Therefore option (3) is the correct answer.

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.

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