MCQ Test: Linear Inequalities - Bank Exams MCQ

Given,

⇒ x² - 6x - 27 > 0

⇒ x² - 9x + 3x - 27 > 0

⇒ x(x - 9) + 3(x - 9) > 0

⇒ (x - 9)(x + 3) > 0

As we know, when ab > 0, then there are two cases:

⇒ Either a > 0 and b > 0

⇒ Or a < 0 and b < 0

Considering the first case,

⇒ (x - 9) > 0 and (x + 3) > 0

⇒ x > 9 and x > - 3

⇒ x > 9

Considering the second case,

⇒ (x - 9) < 0 and (x + 3) < 0

⇒ x < 9 and x < - 3

⇒ x < - 3

∴ x < - 3 or x > 9 

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: 2(3x - 4) -2 < 4x - 2 ≥ 2x – 4

First by solving the inequation: 2(3x - 4) -2 < 4x - 2 we get

⇒ 6x - 10 < 4x - 2

⇒ 2x < 8

⇒ x < 4--------(1)

Similarly, by solving the inequation 4x - 2 ≥ 2x - 4 we get

⇒ 2x ≥ - 2

⇒ x ≥ - 1--------(2)

From equation (1) and (2) we can say that -1 ≤ x < 4

So, out of the given options the possible value which x can take is 2.

Given:

Equation₁ = ax + 9y = 1

Equation₂ = 9y - x - 1 = 0 

Concept used:

If linear equations are a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Here, the equations have an infinite number of solutions, if

a₁/a₂ = b₁/b₂ = c₁/c₂

Calculation:

We have equations,

ax + 9y = 1 

⇒ ax + 9y - 1 = 0

and, 9y - x - 1 = 0

⇒ x - 9y + 1 = 0

Here, a₁ = a, b₁ = 9, c₁ = -1

and, a₂ = 1, b₂ = -9, c₂ = 1

As we know that 

a₁/a₂ = b₁/b₂ = c₁/c₂

⇒ a/1 = 9/-9 = -1/1

⇒ a = -1 = -1

⇒ a = -1

∴ The value of a is -1.

Calculation: 

Given: 9a - a² ≤ 17a + 15

On rearranging

-a² + 9a ≤ 17a + 15

Shifting the sign

a² - 9a ≥ - 17a - 15

a2 - 9a + 17a + 15 ≥ 0

a² + 8a + 15 ≥ 0

a² + 5a + 3a + 15 ≥ 0

a(a+ 5) + 3(a + 5)  ≥ 0

(a + 3)(a + 5)  ≥ 0

So, -3 and -5 are the roots of the equation.

Now look at the diagram given below,

We see that all the numbers less than -5 and all the numbers greater than -3 will give us positive result. While numbers between -5 and -3 will give us negative results

So, all the above value holds for the above equation.

Given:

Initial ratio = 5 ∶ 6

Final ratio should be less than 17 ∶ 22

Calculation:

Let the least whole number that is needed to be subtracted be a.

According to the question,

(5 - a)/(6 - a) < 17/22

⇒ 5 × 22 - 22a < 17 × 6 - 17a 

⇒ 110 - 22a < 102 - 17a 

⇒ 110 - 102 < - 17a + 22a 

⇒ 8 < 5a 

⇒ 8/5 = 1.6 < a 

∴ The least whole number must be 2.

Calculation:

From Statement I.

x² - 5x + 6 = 0

(x - 2) (x - 3) = 0

x = 2, 3

From Statement II.

y² - 9y + 20 = 0

(y - 5) (y - 4) = 0

y = 5, 4

Hence x < y

⇒ 6x + 2(6 - x) > 2x - 2

⇒ 6x + 12 - 2x > 2x - 2

⇒ 2x > - 14

⇒ x > - 7     ----(1)

⇒ 2x - 2 < 5x/2 - 3x/4

⇒ 2x - 2 < 7x/4

⇒ 8x - 7x < 8

⇒ x < 8     ----(2)

From (1) and (2),

- 7 < x < 8

∴ x = 5 satisfies the given conditions from the above options.

Calculation-

⇒ -2 < x - 4 < 2
⇒ 2 < x < 6 
As x ≠ 4 hence x ϵ (2, 4) ∪ (4, 6) 
Since, (4, 6) is lying in the interval for all the acceptable values of x ϵ (2, 4) ∪ (4, 6) so this will be the best option.  
∴ The solution of equality  > 1 is x ϵ (4, 6).

-4 = -7 + 3x
⇒ 3x = 7 – 4
⇒ x = 3/3
∴ x = 1 

a + b = 5      ----(1)

2a – b = 4      ----(2)

Adding Eq. (1) and (2), we get

⇒ a + b + 2a – b = 5 + 4

⇒ 3a = 9

⇒ a = 3

Putting a = 3 in eq. (1) we get b = 2

∴ a > b

2x + 5 > 2 + 3x

5 – 2 > 3x – 2x

3 > x _____(1)

2x - 3 ≤ 4x - 5

5 – 3 ≤ 4x – 2x

1 ≤ x ______ (2)

From (1) and (2)

x = 1 or 2

Given:

 x = 8 – 2√(15)

Calculation:

 x = (√5)² + (√3)² – 2√(15)

⇒ x = (√5 - √3)²

⇒ √x = √5 - √3

And 1/√x = (√5 + √3)/2

According to the question,

Given:

a² - b² = 88

a - b = 4

Formula used:

a² - b² = (a - b)(a + b)

Calculation:

a - b = 4      ----(1)

(a - b)(a + b) = 88

⇒ 4 × (a + b) = 88

⇒ a + b = 88/4

⇒ a + b = 22      ----(2)

Adding equation (1) and equation (2), we get

⇒ a - b + a + b = 4 + 22

2a = 26

⇒ a = 13

Put the value of a in equation (2), we get

13 + b = 22

⇒ b = 9

value of ab = 13 × 9

⇒ ab = 117

∴ The value of ab is 117.

3x + 4(1 – x) > 5x – 2

3x + 4 – 4x > 5x – 2

6 > 6x

x < 1 - - - - - - - - - - - - - - (1)

5x – 2 > 3x – 4

2x > -2

x > -1 - - - - - - - - - - - - - - (2)

From (1) and (2)

x = 0

4(x + 5) – 3 > 6 – 4x ≥ x – 5

4x + 20 – 3 > 6 – 4x ≥ x – 5

4x + 17 > 6 – 4x ≥ x – 5

Adding 4x in the above equation

8x + 17 > 6 ≥ 5x – 5

So, x > -11/8 and 11/5 ≥ x

Therefore, based on options x = 2