Solve the inequality 2x - 3 < x + 2 ≤ 3x + 5. show your work.

-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.

step-by-step explanation:

abc