Question 1 of 40intermediate🔧 ApplyNumerical2 marks

Solve the inequality: 2≤3x−4≤52 \leq 3x - 4 \leq 5

Correct Answer

The solution set of the inequality is x∈[2,3]x \in [2, 3].

Solution:

We have 2≤3x−4≤52 \leq 3x - 4 \leq 5

or 2+4≤3x−4+4≤5+42 + 4 \leq 3x - 4 + 4 \leq 5 + 4 (adding 4 throughout)

or 6≤3x≤96 \leq 3x \leq 9

or 63≤3x3≤93\frac{6}{3} \leq \frac{3x}{3} \leq \frac{9}{3} (dividing by 3 throughout)

or 2≤x≤32 \leq x \leq 3

Thus, all real numbers xx which are greater than or equal to 2 and less than or equal to 3 are the solutions of the given inequality. Therefore, the solution set is [2,3][2, 3].

Exercise: Miscellaneous Exercise on Chapter 5 | Q: 1 | (Chapter: Page 10)
For More Understanding

Explanation

This question follows the pattern of Example 9 and Example 10 from the textbook context, where compound inequalities are solved simultaneously. The key method is to perform the same operation on all three parts of the inequality. Adding 4 to all parts isolates the term with x, then dividing by 3 gives the final solution range.

Solution Steps

  1. Step 1: Add 4 to all parts: 2+4≤3x−4+4≤5+42 + 4 \leq 3x - 4 + 4 \leq 5 + 4, giving 6≤3x≤96 \leq 3x \leq 9

  2. Step 2: Divide all parts by 3: 63≤3x3≤93\frac{6}{3} \leq \frac{3x}{3} \leq \frac{9}{3}, giving 2≤x≤32 \leq x \leq 3

  3. Step 3: Write the solution set: x∈[2,3]x \in [2, 3]

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