Question 13 of 40intermediate🔧 ApplyNumerical3 marks

Solve the inequality for real xx: x4<(5x−2)3−(7x−3)5\frac{x}{4} < \frac{(5x - 2)}{3} - \frac{(7x - 3)}{5}

Correct Answer

x ∈ (4, ∞)

Exercise: EXERCISE 5.1 | Q: 15 | (Chapter: Page 7)
For More Understanding

Explanation

This question involves solving a linear inequality with fractions. Following the method shown in Example 4 and Example 6 of the textbook, we first eliminate the denominators by multiplying throughout by the LCM of all denominators. The key steps involve: (1) finding LCM of 4, 3, and 5 which is 60, (2) multiplying all terms by 60 to eliminate fractions, (3) simplifying and collecting like terms, and (4) applying Rule 2 - when dividing by a negative number, the inequality sign reverses. The final solution is all real numbers greater than 4.

Solution Steps

  1. Step 1: Given inequality: x/4 < (5x - 2)/3 - (7x - 3)/5

  2. Step 2: Find LCM of denominators 4, 3, 5 = 60. Multiply both sides by 60: 60(x/4) < 60[(5x - 2)/3] - 60[(7x - 3)/5]

  3. Step 3: Simplify: 15x < 20(5x - 2) - 12(7x - 3)

  4. Step 4: Expand: 15x < 100x - 40 - 84x + 36

  5. Step 5: Simplify: 15x < 16x - 4

  6. Step 6: Rearrange: 15x - 16x < -4, which gives -x < -4

  7. Step 7: Multiply both sides by -1 (sign reverses by Rule 2): x > 4

  8. Step 8: Solution set: x ∈ (4, ∞)

PreviousNext