Question 16 of 40intermediate🔧 ApplyNumerical3 marks

Solve the inequality for real xx: (2x−1)3≥(3x−2)4−(2−x)5\frac{(2x - 1)}{3} \geq \frac{(3x - 2)}{4} - \frac{(2 - x)}{5}

Correct Answer

x ≤\leq 2, i.e., x ∈ (-∞, 2]

Exercise: EXERCISE 5.1 | Q: 16 | (Page Page 7)
For More Understanding

Explanation

This question tests solving linear inequalities with fractions. Following the method shown in Example 4 and Example 6 from the textbook, we eliminate fractions by multiplying throughout by the LCM of denominators, then solve for x. The key point is remembering Rule 2: when dividing by a negative number, the inequality sign reverses.

Solution Steps

  1. Step 1: Find LCM of denominators 3, 4, and 5. LCM = 60.

  2. Step 2: Multiply both sides by 60: 20(2x - 1) ≥\geq 15(3x - 2) - 12(2 - x)

  3. Step 3: Expand all terms: 40x - 20 ≥\geq 45x - 30 - 24 + 12x

  4. Step 4: Simplify: 40x - 20 ≥\geq 57x - 54

  5. Step 5: Rearrange: -17x ≥\geq -34

  6. Step 6: Divide by -17 and reverse inequality sign (Rule 2): x ≤\leq 2

  7. Step 7: Solution set is (-∞, 2]

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