Answer:
-3/2 ≤ x < 5
Step-by-step explanation:
Begin the solution of this inequality 2x - 3 < x + 2 ≤ 3x + 5 by combining the constants: Subtracting 2 from 2x - 3, x + 2 and 3x + 5 yields:
2x - 5 < x ≤ 3x + 3.
Focus on the first part of this inequality, 2x - 5 < x simplifies to x - 5 < 0 if we subtract x from both sides. Then x < 5 is part of the solution.
Focus now on the second part of this inequality: x ≤ 3x + 3. Subtracting x from both sides yields 0 ≤ 2x + 3, or -3 ≤ 2x, or 2x ≥ -3, or x ≥ -3/2.
Re-combining these two separate inequalities results in:
-3/2 ≤ x < 5 (equivalent to -1.5 ≤ x < 5).
Example: suppose we choose a test number from this interval. Does this test number satisfy the original equation? Choose x = 1.
Is 2(1) - 3 < 1 + 2 ≤ 3(1) + 5 true?
Is 2 - 3 < 3 ≤ 8 true? YES. So x = 1 is part of the solution set -3/2 ≤ x < 5.
