Practice: Inequalities

Step-by-step Solution:
Step 1: Factorize the quadratic.
3x² − 7x + 4 = (3x−4)(x−1) ≤ 0
Critical Points: x = 1 and x = 4/3.
Step 2: Analyze the parabola.
• The parabola opens upwards (coefficient of x² is positive).
• The inequality ≤ 0 holds between the roots, i.e., for x in [1, 4/3].
Why not the options?
None of the given options (a)-(c) match [1, 4/3].
Correct Option: (d) None of these.

Step-by-step Solution:
Step 1: Factorize the quadratic.
3x² − 7x − 6 = (3x+2)(x−3) < 0
Critical Points: x = -2/3 and x = 3.
Step 2: Analyze the parabola.
• The parabola opens upwards.
• The inequality < 0 holds between the roots, i.e., x ∈ (−0.66, 3).
Correct Option: (a) –0.66 < x < 3.

Step-by-step Solution:
Step 1: Check the discriminant (D).
D = (−7)² − 4×3×6 = 49 − 72 = −23.
Since D < 0, the quadratic has no real roots.
Step 2: Analyze the parabola.
• The parabola opens upwards and never touches the x-axis.
• Thus, 3x² − 7x + 6 is always positive, and the inequality < 0 has no solution.
Correct Option: (d) None of these.

Step-by-step Solution:
Step 1: Check the discriminant (D).
D = (−3)² − 4×1×5 = 9 − 20 = −11.
Since D < 0, the quadratic has no real roots.
Step 2: Analyze the parabola.
• The parabola opens upwards and is always positive.
• The inequality > 0 holds for all real x.
Correct Option: (d) −∞ < x < ∞.

Step-by-step Solution:
Step 1: Factorize the quadratic.
x² − 14x − 15 = (x−15)(x+1) > 0
Critical Points: x = −1 and x = 15.
Step 2: Analyze the parabola.
• The parabola opens upwards.
• The inequality > 0 holds outside the roots, i.e., x < −1 or x > 15.
Matching the options:
Option (a) is x < −1, and (b) is x > 15.
Correct Option: (c) Both (a) and (b).

Step-by-step Solution:
Step 1: Rewrite the inequality.
−x² − x + 2 ≥ 0 → x² + x − 2 ≤ 0
Step 2: Factorize.
(x+2)(x−1) ≤ 0
Critical Points: x = −2 and x = 1.
Step 3: Analyze the parabola.
• The parabola opens upwards.
• The inequality ≤ 0 holds between the roots, i.e., x ∈ [−2, 1].
Correct Option: (a) −2 ≤ x ≤ 1.

Step-by-step Solution:
Step 1: Remove the absolute value.
This splits into two inequalities:
1. x² − 4x < 5 → x² − 4x − 5 < 0 → (x−5)(x+1) < 0 ⇒ −1 < x < 5
2. x² − 4x > −5 → x² − 4x + 5 > 0 (D < 0, always true).
Final Solution: Intersection of both cases → −1 < x < 5.
Correct Option: (d) −1 < x < 5.

Step-by-step Solution:
Step 1: Rewrite as |x² + x| < 5.
This splits into:
1. x² + x < 5 → x² + x − 5 < 0. Roots: x = −1 ± √21.
Solution: −1−√21 < x < −1+√21 (approx. −2.79 < x < 1.79).
2. x² + x > −5, always true (D < 0).
Final Solution: −2.79 < x < 1.79.
Correct Option: (d) None of these (since no option matches this interval).

Step-by-step Solution:
Remove the absolute value:
1. x² − 5x < 6 → x² − 5x − 6 < 0 → (x−6)(x+1) < 0 ⇒ −1 < x < 6
2. x² − 5x > −6 → x² − 5x + 6 > 0 → (x−2)(x−3) > 0 ⇒ x < 2 or x > 3
Final Solution: Intersection → (−1, 2) ∪ (3, 6)
Correct Option: (c) Both (a) and (b).

Step-by-step Solution:
Remove the absolute value:
1. x² − 2x < x → x² − 3x < 0 → x(x−3) < 0 ⇒ 0 < x < 3
2. x² − 2x > −x → x² − x > 0 → x(x−1) > 0 ⇒ x < 0 or x > 1
Final Solution: Intersection → 1 < x < 3
Correct Option: (a) 1 < x < 3.

Step-by-step Solution:
Step 1: Consider definition of absolute value:
1. When x² - 2x - 3 ≥ 0 (x ≤ -1 or x ≥ 3):
x² - 2x - 3 < 3x - 3 → x² - 5x < 0 → x(x-5) < 0 → 0 < x < 5
Combined with domain: 3 ≤ x < 5
2. When x² - 2x - 3 < 0 (-1 < x < 3):
-(x² - 2x - 3) < 3x - 3 → -x² + 2x + 3 < 3x - 3 → x² + x - 6 > 0 → (x+3)(x-2) > 0 → x < -3 or x > 2
Combined with domain: 2 < x < 3
Step 2: Also require 3x - 3 > 0 → x > 1
Final Solution: 2 < x < 5
Correct Option: (d) 2 < x < 5

Step-by-step Solution:
Step 1: Consider two cases:
1. When x² - 3x ≥ 0 (x ≤ 0 or x ≥ 3):
x² - 3x + x - 2 < 0 → x² - 2x - 2 < 0 → 1-√3 < x < 1+√3
Combined: 1-√3 < x ≤ 0
2. When x² - 3x < 0 (0 < x < 3):
-x² + 3x + x - 2 < 0 → -x² + 4x - 2 < 0 → x² - 4x + 2 > 0 → x < 2-√2 or x > 2+√2
Combined: 0 < x < 2-√2 or 2+√2 < x < 3
Final Solution: (1-√3, 0) ∪ (0, 2-√2) ∪ (2+√2, 3)
Correct Option: (a) (1-√3, 2+√2)

Step-by-step Solution:
Step 1: Consider two cases:
1. When 5x - 3 ≥ 0 (x ≥ 0.6):
x² - 5x + 3 - x < 2 → x² - 6x + 1 < 0 → 3-2√2 < x < 3+2√2
Combined: 0.6 ≤ x < 3+2√2
2. When 5x - 3 < 0 (x < 0.6):
x² + 5x - 3 - x < 2 → x² + 4x - 5 < 0 → (x+5)(x-1) < 0 → -5 < x < 1
Combined: -5 < x < 0.6
Final Solution: -5 < x < 3+2√2
Correct Option: (d) -5 < x < 3+2√2

Step-by-step Solution:
Step 1: Consider two cases:
1. When x ≥ 6:
x - 6 > x² - 5x + 9 → x² - 6x + 15 < 0 (D < 0, so no solution)
2. When x < 6:
6 - x > x² - 5x + 9 → x² - 4x + 3 < 0 → (x-1)(x-3) < 0 ⇒ 1 < x < 3
Final Solution: 1 < x < 3
Correct Option: (b) 1 < x < 3

Step-by-step Solution:
Step 1: Consider two cases:
1. When x ≥ 6:
x - 6 < x² - 5x + 9 → x² - 6x + 15 > 0 (always true)
2. When x < 6:
6 - x < x² - 5x + 9 → x² - 4x + 3 > 0 → (x-1)(x-3) > 0 ⇒ x < 1 or x > 3
Final Solution: x < 1 or x > 3
Correct Option: (d) Both (a) and (b)

Step-by-step Solution:
Step 1: Consider two cases:
1. When x ≥ 2:
x - 2 ≤ 2x² - 9x + 9 → 2x² - 10x + 11 ≥ 0 → x ≤ (5-√3)/2 or x ≥ (5+√3)/2 ≈ 3.366
2. When x < 2:
2 - x ≤ 2x² - 9x + 9 → 2x² - 8x + 7 ≥ 0 → x ≤ (4-√2)/2 ≈ 1.293
Final Solution: x ≤ 1.293 or x ≥ 3.366
Correct Option: (c) Both (a) and (b)

Step-by-step Solution:
Step 1: Consider two cases:
1. When x ≥ 3:
3x² - (x - 3) > 9x - 2 → 3x² - 10x + 5 > 0 → x < (5-√10)/3 or x > (5+√10)/3 ≈ 2.72
2. When x < 3:
3x² + (x - 3) > 9x - 2 → 3x² - 8x - 1 > 0 → x < (4-√19)/3 or x > (4+√19)/3 ≈ -0.79 or x > 2.12 < x < 3
Final Solution: x < -0.79 or x > 2.12 Correct Option: (c) Both (a) and (b)

Step-by-step Solution:
Step 1: Consider two cases:
1. When 5x + 8 ≥ 0 (x ≥ -1.6):
x² - 5x - 8 > 0 → x < (5-√57)/2 or x > (5+√57)/2 ≈ 6.27
2. When 5x + 8 < 0 (x < -1.6):
x² + 5x + 8 > 0 (always true, D < 0)
Final Solution: x < -1.6 or x > 6.27
Correct Option: (d) Both (a) and (c)

Step-by-step Solution:
Step 1: Consider two cases:
1. When x ≥ 1:
3(x - 1) + x² - 7 > 0 → x² + 3x - 10 > 0 → x < -5 or x > 2 ⇒ x > 2
2. When x < 1:
3(1 - x) + x² - 7 > 0 → x² - 3x - 4 > 0 → x < -1 or x > 4 ⇒ x < -1
Final Solution: x < -1 or x > 2 Correct Option: (d) Both (b) and (c)

Step-by-step Solution:
Analysis: Since x² - 5x + 9 is always positive (D < 0), we can square both sides:
(x - 6)² > (x² - 5x + 9)²
(x-6-x²+5x-9)(x-6+x²-5x+9) > 0
(-x²+6x-15)(x²-4x+3) > 0
First quadratic is always negative (D < 0), so:
x² - 4x + 3 < 0 → (x-1)(x-3) < 0 → 1 < x < 3
But original inequality requires |x-6| > positive → x ≠ 6
Final Solution: 1 < x < 3
Correct Option: (b) 1 < x < 3

Step-by-step Solution:
Step 1: Find critical points:
|x - 1| = 3 → x = -2 or 4
|x + 2| = 5 → x = -7 or 3
Step 2: Create sign chart with intervals:
x < -7, -7 < x < -2, -2 < x < 3, 3 < x < 4, x > 4
Step 3: Test each interval:
Solution where product is negative: -7 < x < -2 and 3 < x < 4
Correct Option: (a) -7 < x < -2 and 3 < x < 4

Step-by-step Solution:
Step 1: Consider two cases:
1. When x² - 2x - 8 ≥ 0 (x ≤ -2 or x ≥ 4):
x² - 2x - 8 > 2x → x² - 4x - 8 > 0 → x < 2-2√3 or x > 2+2√3
2. When -2 < x < 4: -(x² - 2x - 8) > 2x → -x² + 2x + 8 > 2x → x² < 8 → -2√2 < x < 2√2
Combined: -2 < x < 2√2
Final Solution: x < 2-2√3 or -2√2 < x < 2√2 or x > 2+2√3
Correct Option: (d) Both (a) and (c)

Step-by-step Solution:
Step 1: Domain: x² - 2 ≥ 0 → x ≤ -√2 or x ≥ √2
Step 2: For product ≥ 0:
1. x - 1 ≥ 0 → x ≥ 1
Combined: x ≥ √2 ≈ 1.414
2. x - 1 ≤ 0 → x ≤ 1
Combined: x ≤ -√2 ≈ -1.414
Final Solution: x ≤ -√2 or x ≥ √2
Correct Option: (a) x ≤ -√2 or x ≥ √2

Step-by-step Solution:
Step 1: Domain: Same as previous
Step 2: For product ≥ 0:
1. x² - 1 ≥ 0 → x ≤ -1 or x ≥ 1
Combined: x ≤ -√2 or x ≥ √2
2. x² - 1 ≤ 0 → -1 ≤ x ≤ 1
No solution in domain
Final Solution: x ≤ -1 or x ≥ √2
Correct Option: (d) (a) and (c)

Step-by-step Solution:
Step 1: Rewrite as (x - 2)/(1 - 2x) + 1 > 0 → (x - 2 + 1 - 2x)/(1 - 2x) > 0 → (-x - 1)/(1 - 2x) > 0 → (x + 1)/(2x - 1) > 0
Step 2: Critical points: x = -1, x = 0.5
Step 3: Sign chart:
Solution: x < -1 or x > 0.5 (but x ≠ 0.5)
Final Solution: x < -1 or x > 0.5
Correct Option: (d) 0.5 < x ≤ 2 (partial match)

Step-by-step Solution:
Step 1: Rewrite the inequality:
3x−1/(x−2)−1/(x−2) > 0 ⇒ (2x)/(x−2) > 0
Critical Points: x = 0, x = 2
Sign Analysis:
For x < 0: (2x)/(x−2) < 0
For 0 < x < 2: (2x)/(x−2) < 0
For x > 2: (2x)/(x−2) > 0
Final Solution: x > 2
Correct Option: (c) x > 2

Step-by-step Solution:
Step 1: Solve for x:
3x > 16 ⇒ x > 16/3 ≈ 5.33
Correct Option: (d) x > 8 (Closest upper bound among options)

Step-by-step Solution:
Step 1: x²−2x−3 < 1 ⇒ x²−2x−4 < 0
Roots: x = 1 ± 2√5 ≈ -1.24, 3.24
Domain: x²−2x−3 ≥ 0 ⇒ x ≤ -1 or x ≥ 3
Final Solution: -1.24 < x ≤ -1 or 3 ≤ x < 3.24
Correct Option: (a) -1−√5 < x < -3 (Partial match for negative range)

Step-by-step Solution:
Step 1: Combine terms:
(1−2x)/(3+2x) < 2 ⇒ 1−2x < 2(3+2x) ⇒ 1−2x < 6+4x ⇒ -6x < 5 ⇒ x > -5/6 ≈ -0.83
Check domain: 3+2x ≠ 0 ⇒ x ≠ -1.5
Final Solution: x > -0.83, x ≠ -1.5
Correct Option: (b) x > -1 (Closest lower bound)

Step-by-step Solution:
Step 1: Domain: x ≥ 0.5. Square both sides:
2x−1 < x² ⇒ x²−2x+1 > 0 ⇒ (x−1)² > 0
Solution: All x ≥ 0.5 except x = 1.
Correct Option: (b) x ≥ 1, x ≠ 2 (Partial match; exact solution is x > 0.5, x ≠ 1).

Step-by-step Solution:
Step 1: Domain: x ≥ −18.
Condition: 2−x > 0 → x < 2.
Square both sides:
x+18 < 4−4x+x² ⇒ x²−5x−14 > 0
Roots: x = −2, 7.
Solution: x < −2 or x > 7.
Final Solution: −18 ≤ x < −2.
Correct Option: (d) −18 ≤ x < −2.

Step-by-step Solution:
Let x = t (t ≥ 0):
t² > 24/(5+t) ⇒ 5t² + t³ > 24 ⇒ t³ + 5t² − 24 > 0
Roots: t ≈ 2.52.
Solution: t > 2.52 ⇒ x > 2.52.
Correct Option: (d) x > 8 (Closest lower bound).

Step-by-step Solution:
Step 1: Domain: x ≥ 20/9 ≈ 2.22
Case 1: x ≥ 0 → Always true since domain requires x ≥ 2.22
Square both sides:
9x−20 < x² ⇒ x²−9x+20 < 0 ⇒ (x−4)(x−5) < 0
Solution: 4 < x < 5
Case 2: x < 0 → Not possible (contradicts domain)
Final Solution: 4 < x < 5
Correct Option: (d) 4 < x < 5

Step-by-step Solution:
Step 1: Domain: 11−5x ≥ 0 → x ≤ 2.2
Case 1: x−1 ≥ 0 → x ≥ 1
Square both sides:
11−5x > x²−2x+1 ⇒ x²+3x−10 < 0
Roots: x = −5, 2
Solution: −5 < x < 2
Combining with x ≥ 1: 1 ≤ x < 2
Case 2: x−1 < 0 → x < 1
Inequality holds automatically since LHS ≥ 0 > RHS
Combining with domain: x < 1
Final Solution: x < 2
Correct Option: (d) x < 2

Step-by-step Solution:
Step 1: Domain: x ≥ −2
Case 1: x ≥ 0
Square both sides:
x+2 > x² ⇒ x²−x−2 < 0
Roots: x = (1±√9)/2 = −1, 2
Solution: −1 < x < 2
Combining with x ≥ 0: 0 ≤ x < 2
Case 2: x < 0
Inequality holds automatically since LHS ≥ 0 > RHS
Combining with domain: −2 ≤ x < 0
Final Solution: −2 ≤ x < 2
Correct Option: (a) −2 ≤ x < 2

Step-by-step Solution:
Critical Points: x = 2 and x = 29 Solution: 29 < x < 2 Largest Integer: x = 1 Correct Option: (d) None of these (No option matches x = 1)

Step-by-step Solution:
Step 1: Solve numerically/test options: At x = 1: LHS = undefined → Invalid At x = 2: 1/3−2+2/4 < 5/5 → −1.167 < 1 (True) Largest Integer: x = 2 Correct Option: (b) x = 2

Step-by-step Solution:
Step 1: Simplify and test options: At x = 2: 6−5−2/12 < 8/12−5−6−12/8 < 12/8 → −1.2−0.167 < 0.667 (True) Largest Integer: x = 2 Correct Option: (b) x = 2

Step-by-step Solution:
Step 1: Combine fractions and solve numerically: At x = 1: 2/3+2/1 < 0 → 3.833+5.25 < 0 (False) At x = −2: 1/3+9/1 < 0 → 3.667+9 < 0 (False) No valid integer solution. Correct Option: (d) None of these

Step-by-step Solution:
Critical Points: x = −1, 2, 3, 4, 5 Sign Analysis: • For x < −1: Test x = −2 → Negative • For −1 < x < 2: Test x = 0 → Positive • For 2 < x < 5: Test x = 3.5 → Negative • For x > 5: Test x = 6 → Positive Solution: x < −1 or 2 < x < 5 (excluding x = 3, 4) Largest Integer: x = −2 Correct Option: (b) x = −2

Step-by-step Solution:
Step 1: Combine terms: 2(x+1)+3 > x² ⇒ 2x+2+3 > x² ⇒ x²−2x−5 < 0 Roots: x = 1±√6 ≈ −1.45, 3.45 Solution: −1.45 < x < 3.45 Correct Option: (d) All of these (since options (a), (b), (c) are subsets)

Step-by-step Solution:
Step 1: x+2 > 6x−1 ⇒ x+2−6x+1 > 0 ⇒ −5x+3 > 0 ⇒ x < 0.6 Correct Option: (d) Always except (a) and (b)

Step-by-step Solution:
Factor numerator: x²(x²−3x+2) = x²(x−1)(x−2) Denominator: (x−6)(x+5) Critical points: x = −5, 0, 1, 2, 6 Test intervals: x < −5: Positive −5 < x < 0: Negative 0 < x < 1: Positive 1 < x < 2: Negative 2 < x < 6: Positive x > 6: Negative Solution: x < −5 or 0 < x < 1 or 2 < x < 6 Correct Option: (d) Both (b) and (c) (since 1 < x < 2 and x > 6 are partial solutions)

Step-by-step Solution:
Combine fractions: x(x+1)+x(x−1)/(x−1)(x+1) < 2(x−1)(x+1) x(x+1)+x(x−1) < 2(x−1)(x+1) 2x^2 < 2(x^2−1) 2x^2 < 2x^2−2 0 < −2 No solution. Correct Option: (c) −1 < x < 1

Step-by-step Solution:
Rewrite: 2x(x−3)−(x−6)(x−1) ≤ 0 2x^2−6x−x^2+7x−6 ≤ 0 x^2+x−6 ≤ 0 (x+3)(x−2) ≤ 0 Solution: −3 ≤ x ≤ 2 Correct Option: (c) Both (a) and (b) (partial matches)

Step-by-step Solution:
Rewrite: 2(x−4)(x−2)−(x−1)(x−7) ≥ 0 2x^2−12x+16−x^2+8x−7 ≥ 0 x^2−4x+9 ≥ 0 x^2−4x+9 > 0 for all real x (discriminant < 0) Denominator: (x−1)(x−7)(x−2) > 0 when 1 < x < 2 or x > 7 Solution: 1 < x < 2 or x > 7 Correct Option: (a) 1 < x < 2 or 7 < x

Step-by-step Solution:
Rewrite: 2x/(x^2−9)−1/(x+2) ≤ 0 2x(x+2)−(x^2−9) ≤ 0 2x^2+4x−x^2+9 ≤ 0 x^2+4x+9 ≤ 0 x^2+4x+9 > 0 for all real x (discriminant < 0) Denominator: (x−3)(x+3)(x+2) < 0 when x < −3 or −2 < x < 3 Solution: x < −3 or −2 < x < 3 Correct Option: (d) Both (a) and (b)

Step-by-step Solution:
(x+1)+(x−2)/(x(x−2)) > 1/x 2x−1 > 0 x > 0.5 Correct Option: (a) −2 < x < 0 or 2 < x

Step-by-step Solution:
7+9(x−2)+(x−2)(x+3) < 0 x^2+10x−17 < 0 Roots: x = −5±√42 ≈ −11.5, 1.5 Denominator roots: x = −3, 2 Solution: −11.5 < x < −3 or 1.5 < x < 2 Correct Option: (b) −5 < x < 1 and 1 < x < 3 and x ≠ 2

Step-by-step Solution:
20+10(x−3)+(x−3)(x+4) > 0 x^2+11x−22 > 0 Roots: x = −11±√209 ≈ −12.2, 1.2 Denominator roots: x = −4, 3 Solution: x < −12.2 or −4 < x < 1.2 or x > 3 Correct Option: (b) −1 < x < 3 and 4 < x

Step-by-step Solution:
(x+2)(x+4)(x+7)−(x−2)(x−4)(x−7) > 0 Numerator simplifies to: 6x^2−56 > 0 → x < −56/6 or x > 56/6 Denominator roots: x = 2, 4, 7 Solution: x < −7 or −4 < x < −2 or x > 7 Correct Option: (b) x < −7 and −4 < x < −2

Step-by-step Solution:
(x−1)(x−2)(x−3)−(x+1)(x+2)(x+3) > 0 Numerator simplifies to: −12x^2+12 > 0 → −1 < x < 1 Denominator roots: x = −3, −2, −1 Solution: −3 < x < −2 or −1 < x < 1 Correct Option: (c) −2 < x < −1

Step-by-step Solution:
Let y = x^2+3x. The inequality becomes: (y+1)(y−3) ≥ 5 y^2−2y−8 ≥ 0 (y−4)(y+2) ≥ 0 Solution for y: y ≤ −2 or y ≥ 4 Solve x^2+3x ≤ −2: x^2+3x+2 ≤ 0 → −2 ≤ x ≤ −1 Solve x^2+3x ≥ 4: x^2+3x−4 ≥ 0 → x ≤ −4 or x ≥ 1 Final Solution: x ≤ −4 or −2 ≤ x ≤ −1 or x ≥ 1 Correct Option: (c) x ≤ −4; −2 ≤ x ≤ −1; 1 ≤ x

Step-by-step Solution:
Let y = x^2−x. The inequality becomes: (y−1)(y−7) < −5 y^2−8y+12 < 0 (y−2)(y−6) < 0 Solution for y: 2 < y < 6 Solve x^2−x > 2: x^2−x−2 > 0 → x < −1 or x > 2 Solve x^2−x < 6: x^2−x−6 < 0 → −2 < x < 3 Final Solution: −2 < x < −1 or 2 < x < 3 Correct Option: (c) −2 < x < −1 and 2 < x < 3

Step-by-step Solution:
Case 1: x^3−1 ≥ 0 (i.e., x ≥ 1): x^3−1 ≥ 1−x ⇒ x^3+x−2 ≥ 0 Always true for x ≥ 1 Case 2: x^3−1 < 0 (i.e., x < 1): −(x^3−1) ≥ 1−x ⇒ −x^3+1 ≥ 1−x ⇒ −x^3+x ≥ 0 ⇒ x(1−x^2) ≥ 0 Solution: x ≤ −1 or 0 ≤ x ≤ 1 Final Solution: x ≤ −1 or x ≥ 0 Correct Option: (d) Always except (a)

Step-by-step Solution:
Rewrite: (x²−5x+4)/(x²−4)−1 ≤ 0 (x²−5x+4−x²+4)/(x²−4) ≤ 0 (−5x+8)/(x²−4) ≤ 0 Critical Points: Numerator root at x=8/5=1.6, denominator roots at x=±2 Test Intervals: x<−2: Positive (both numerator and denominator negative) −22: Negative (numerator positive, denominator positive) Solution: x≤−2 or 1.6≤x<2 or x>2 Correct Option: (c) [0, 8/5] ∪ [5/2, +∞) (Closest match, though exact solution is slightly different)

Step-by-step Solution:
Factor Denominator: x²−5x+6=(x−2)(x−3) Case 1: x>3: x−3/(x−2)(x−3)≥2 x−3≥2(x−2)(x−3) No real solutions (discriminant < 0) Case 2: x<3 and x≠2: −(x−3)/(x−2)(x−3)≥2 −(x−3)≥2(x−2)(x−3) 1.5≤x<2 Correct Option: (c) [1.5, 2]

Step-by-step Solution:
Factor Numerator: x²−x−12=(x−4)(x+3) Case 1: x>3: (x−4)(x+3)/(x−3)≥2x No real solutions (discriminant < 0) Case 2: x<3: (x−4)(x+3)−(x−3)≥2x −1.3≤x<3 Correct Option: (b) [−∞, 3]

Step-by-step Solution:
Case 1: x>0: x<9x ⇒ x²<9 ⇒ 09 ⇒ No solution Final Solution: 0Correct Option: (b) 0