Solve : 2 x + 5 + x - 1 > 8

Given: 2 x + 5 + x - 1 > 8 Here, 2 x + 5 ≥ 0 ⇒ x ≥ - 5 2 and x - 1 ≥ 0 ⇒ x ≥ 1 . Now, 2 x + 5 + x - 1 > 8 ⇒ 2 x + 5 + x - 1 + 2 2 x + 5 x - 1 > 64 ⇒ 3 x + 4 + 2 2 x + 5 x - 1 > 64 ⇒ 2 2 x + 5 x - 1 > 60 - 3 x ⇒ 2 2 x 2 + 3 x - 5 > 60 - 3 x ⇒ 4 2 x 2 + 3 x - 5 > 3600 + 9 x 2 - 360 x ⇒ 4 x 2 + 12 x - 20 > 3600 + 9 x 2 - 360 x ⇒ x 2 - 372 x + 3620 < 0 ⇒ x - 10 x - 362 < 0 ⇒ x ∈ 10 , 362 Required common solution is x ∈ 10 , 362 .

Solve : x 2 + 3 x + x 2 - 2 ≥ 0

Given: x 2 + 3 x + x 2 - 2 ≥ 0 Case 1 : If x ≥ 0 , then x 2 + 3 x + x 2 - 2 ≥ 0 ⇒ 2 x 2 + 3 x - 2 ≥ 0 ⇒ 2 x 2 + 4 x - x - 2 ≥ 0 ⇒ 2 x - 1 x + 2 ≥ 0 ⇒ x ≤ - 2 and x ≥ 1 2 Hence, x ∈ [ 1 2 , ∞ ) Case 2 : If x ≤ - 3 , then 2 x 2 + 3 x - 2 ≥ 0 ⇒ x ≤ - 2 and x ≥ 1 2 Hence, x ≤ - 3 . Case 3 : If - 3 < x < 0 , then - x 2 - 3 x + x 2 - 2 ≥ 0 ⇒ - 3 x - 2 ≥ 0 ⇒ 3 x ≤ - 2 ⇒ x ≤ - 2 3 Hence, x ∈ ( - 3 , - 2 3 , ] Common solution is x ∈ ( - ∞ - 2 3 , ] ∪ [ 1 2 , ∞ )

Solve: 2 3 x - 1 x - 1 3 < 8 x - 3 3 x - 7

We have, 2 3 x - 1 x - 1 3 < 8 x - 3 3 x - 7 Here, x ≠ 1 , 7 3 2 3 x - 1 x - 1 < 8 x - 3 3 x - 7 3 ⇒ 2 3 x - 1 x - 1 < 2 3 x - 3 3 x - 7 3 ⇒ 2 3 x - 1 x - 1 < 2 9 x - 27 3 x - 7 ⇒ 3 x - 1 x - 1 < 9 x - 27 3 x - 7 ⇒ 9 x - 27 3 x - 7 - 3 x - 1 x - 1 > 0 ⇒ 9 x - 27 x - 1 - 3 x - 1 3 x - 7 3 x - 7 x - 1 > 0 ⇒ 9 x 2 - 36 x + 27 - 9 x 2 - 24 x + 7 3 x - 7 x - 1 > 0 ⇒ - 12 x + 20 3 x - 7 x - 1 > 0 ⇒ - 3 x + 5 3 x - 7 x - 1 > 0 It is possible if - 3 x + 5 > 0 ⇒ x > 5 3 and 3 x - 7 x - 1 > 0 ⇒ x < 1 & x > 7 3 Hence, x > 7 3 Case 2 : If - 3 x + 5 < 0 ⇒ x < 5 3 and 3 x - 7 x - 1 < 0 ⇒ x ∈ 1 , 7 3 Therefore, x ∈ 5 3 , 7 3 So, common solution is x ∈ 5 3 , ∞ - 7 3 .

E M B I B E

Gamma Question Bank for Engineering Mathematics>Class XI>Chapter 3 - Quadratic Equations>EXERCISE 5.3

Embibe Experts Solutions for Exercise 3: EXERCISE 5.3

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Embibe Experts Mathematics Solutions for Exercise - Embibe Experts Solutions for Exercise 3: EXERCISE 5.3

Attempt the free practice questions from Exercise 3: EXERCISE 5.3 with hints and solutions to strengthen your understanding. Gamma Question Bank for Engineering Mathematics solutions are prepared by Experienced Embibe Experts.

Questions from Embibe Experts Solutions for Exercise 3: EXERCISE 5.3 with Hints & Solutions

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 3 - Quadratic Equations>EXERCISE 5.3>Q 23

If $α, β, γ$ are the roots of $x^{3} + 8 = 0,$ then the equation whose roots are $α^{2}, β^{2}$ and $γ^{2}$ is

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 3 - Quadratic Equations>EXERCISE 5.3>Q 26

If $α, β$ are the roots of $a x^{2} + b x + c = 0,$ then the equation whose roots are $2 + α, 2 + β$ is

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 3 - Quadratic Equations>EXERCISE 5.3>Q 28

Let $f (x) = x^{3} + 3 x^{2} + 9 x + 6 \sin x,$ then find the number of real roots of the equation $\frac{1}{x - f (1)} + \frac{2}{x - f (2)} + \frac{3}{x - f (3)} = 0$