Inequalities in one Variable MCQ Quiz - Objective Question with Answer for Inequalities in one Variable - 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).

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:

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 6x + 7 ≤ x - 28

⇒ 6x + 7 - x ≤ x - 28 - x 

⇒ 5x + 7 ≤ -28 

⇒ 5x + 7 - 7 ≤ -28 - 7

⇒  5x ≤ -35

⇒  x ≤ -7

Since x is a natural number (which means x must be a positive integer), there are no natural numbers that satisfy (x ≤ -7).

∴ There is no solution 

∴ Option 1 is correct

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 .

Calculations:

Given,1.5 ≤ x ≤ 4.5

So critical points are x = 1.5 =  or x = 4.5 = 

⇒ 2x - 3 = 0 or 2x - 9 = 0

Changing inequality into equality form

∴ (2x - 3)(2x - 9) = 0

As we can see in the wavy curve value of (2x - 3)(2x - 9) is negative.

(2x - 3)(2x - 9) ≤ 0

Hence option 4 is the correct answer.

Concept:

If |x| ≤ a then - a ≤ x ≤ a

Calculations:

Given , |3x - 5| ≤ 2 

⇒ - 2 ≤ 3x - 5 ≤ 2

⇒ - 2 + 5 ≤ 3x  ≤ 2 + 5 

⇒ 3 ≤ 3x  ≤ 7

⇒ 

Hence, if |3x - 5| ≤ 2 then then 

Concept:

The Modulus Function '| |' is defined as:

Calculation:

We have two cases:

CASE I: If x + 5 ≥ 0, then |x + 5| = x + 5.

⇒ x + 5 ≥ 10

⇒ x ≥ 5

⇒ x ∈ [5, ∞)

CASE II: If x + 5 

⇒ -(x + 5) ≥ 10

⇒ -x - 5 ≥ 10

⇒ -x ≥ 15

⇒ x ≤ -15

⇒ x ∈ (-∞, -15]

∴ x ∈ (-∞, -15] ∪ [5, ∞).

Explanation:

Given system of linear inequality is 

3x - 7 \\\ 11 - 5x \le 1 \end{matrix} \right.\)

When 5 + x > 3x - 7 ⇒ 2x

When 11 - 5x ≤ 1 ⇒ 5x ≥ 10 ⇒ x ≥ 2

On the number line, the solution is represented as below.

 x ∈ n

x ≤ 13

Since x ∈ N, Natural number

x > 0

so x ∈ (1, 2, 3, ...13)

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).

Concept:

Modulus Function:

The function f(x) = |x|, defined as |x| = , is called the Modulus function.

Calculation:

Let f(x) = 

Given f(x) ≥ 0

For f(x) to be defined, x - 2 ≠ 0

⇒ x ≠ 2

⇒ f(x) = 2\\ \frac{-(x-2)}{x-2}, x [ x ≠ 2 ]

⇒ f(x) = 

∴ For f(x) ≥ 0

⇒ x > 2

∴ x ∈ (2, ∞)

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:

Comparison using inequality:

For any two real numbers  and  if  then .

Calculation:

Use the given inequality and proceed as follows:

Therefore, we conclude that .