Question 13 of 40intermediate🔧 ApplyNumerical2 marks

Solve the inequality for real xx: 2(2x+3)−10<6(x−2)2(2x + 3) - 10 < 6(x - 2)

Correct Answer

Solution: We have the inequality 2(2x+3)−10<6(x−2)2(2x + 3) - 10 < 6(x - 2). Expanding the brackets on both sides, we get 4x+6−10<6x−124x + 6 - 10 < 6x - 12. Simplifying the left-hand side, we obtain 4x−4<6x−124x - 4 < 6x - 12.

Rearranging the terms to bring variables to one side and constants to the other, we get:

−4+12<6x−4x-4 + 12 < 6x - 4x 8<2x8 < 2x

Dividing both sides by 2, we get 4<x4 < x, which can be written as x>4x > 4.

Since the question asks for real xx, all real numbers greater than 4 are solutions. Therefore, the solution set is x∈(4,∞)x \in (4, \infty).

Exercise: EXERCISE 5.1 | Q: 13 | (Page Page 7)
For More Understanding

Explanation

The solution follows the standard algebraic procedure for linear inequalities as demonstrated in Example 3 and Example 4 of the provided context. We expand the expressions, group like terms, and solve for x. Since the inequality is strict (<), the solution set is an open interval. The final answer format matches the interval notation used in the textbook examples for real number solutions.

Solution Steps

  1. Step 1: Expand the expression to get 4x+6−10<6x−124x + 6 - 10 < 6x - 12.

  2. Step 2: Simplify the inequality to 4x−4<6x−124x - 4 < 6x - 12.

  3. Step 3: Transpose terms to get 8<2x8 < 2x.

  4. Step 4: Divide by 2 to get x>4x > 4.

  5. Step 5: Write the solution set as (4,∞)(4, \infty).

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