What is the solution to 1-(x-3)/3 ≤ 1/2?

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Discussion Overview

The discussion revolves around solving the inequality \(1 - \frac{x-3}{3} \le \frac{1}{2}\). Participants explore the steps involved in manipulating the inequality and the implications of dividing by negative numbers.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant initiates the solution by multiplying the inequality by the common denominator, 6.
  • Another participant provides a continuation of the solution, arriving at \(x \le \frac{9}{2}\).
  • A subsequent participant questions the implications of dividing both sides of an inequality by a negative number, suggesting a need for caution in such operations.
  • Another participant challenges the initial conclusion by providing a counterexample with \(x = -2\), indicating that it does not satisfy the original inequality.
  • One participant corrects the earlier steps, stating that the inequality should flip when dividing by a negative number, leading to \(x \ge \frac{3}{2}\).
  • A later reply reiterates the correction and emphasizes the need to flip the inequality sign, but introduces further corrections to the steps taken.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correct solution to the inequality, with multiple competing views on the final result and the handling of the inequality sign during manipulations.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the treatment of inequalities, particularly when involving negative numbers.

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$1 - \dfrac{x-3}{3} \le \dfrac{1}{2}$

multiply every term by the common denominator, $6$ ...

$6 - 2(x-3) \le 3$

can you finish it?
 
Yes, thank you

-2x - 6 ≤ -3
-2x ≤ -9
x ≤ 9/2
 
frctl said:
Yes, thank you

-2x - 6 ≤ -3
-2x ≤ -9
x ≤ 9/2

... what happens when you divide both sides of an inequality by a negative number ?
 
frctl said:
Yes, thank you

-2x - 6 ≤ -3
-2x ≤ -9
x ≤ 9/2
If x= -2 then x is certainly less than 9/2= 4 but -2x- 6= 4- 6= -2 is not less than -3 so that can't be right.
 
Correction

6 - 2(x - 3) ≤ 3
-2x - 6 ≤ -3
-2x ≤ -3
flip inequality sign
x ≥ 3/2
 
frctl said:
Correction

6 - 2(x - 3) ≤ 3
-2x - 6 ≤ -3 should be -2x + 6 < -3
-2x ≤ -3
flip inequality sign
x ≥ 3/2

correction again
 

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