7
algebra

Solve: 2x5>3x|2x - 5| > 3x

A

x<1x < 1

B

x<5x < -5

C

x>1x > 1

D

x<1x < 1 or x>5x > 5

Correct Answer: A

Choice A is the correct answer. Split into cases.

  1. Case 1: 2x5>3x5>xx<52x - 5 > 3x \rightarrow -5 > x \rightarrow x < -5
  2. Case 2: 2x5<3x5x<5x<12x - 5 < -3x \rightarrow 5x < 5 \rightarrow x < 1
  3. Combine: The union of x<5x < -5 and x<1x < 1 is x<1x < 1 (since x<5x<-5 is inside x<1x<1).
    • Check x=0x=0: 5>0|-5| > 0 (5>0) True.
    • Check x=6x=-6: 17>18|-17| > -18 (17>-18) True.
    • Check x=2x=2: 1>6|-1| > 6 (1>6) False.

Wait, let's re-evaluate carefully. A>B|A| > B is A>BA > B or A<BA < -B.

  1. 2x5>3xx>5x<52x - 5 > 3x \rightarrow -x > 5 \rightarrow x < -5
  2. 2x5<3x5x<5x<12x - 5 < -3x \rightarrow 5x < 5 \rightarrow x < 1 Union: x<1x < 1 OR x<5x < -5. The set x<1x < 1 includes x<5x < -5. So x<1x < 1.

Let's check x=0.9x=0.9: 1.85=3.2>2.7|1.8-5| = 3.2 > 2.7. True. Let's check x=10x=-10: 25=25>30|-25| = 25 > -30. True.

So x<1x < 1 seems correct.

Choice A is correct.Choice B is incorrect (subset). Choice C is incorrect. Choice D is incorrect.