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

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SUMMARY

The solution to the inequality \(1 - \frac{x-3}{3} \le \frac{1}{2}\) is derived through a series of algebraic manipulations. By multiplying each term by the common denominator of 6, the inequality simplifies to \(6 - 2(x-3) \le 3\). This leads to the conclusion that \(x \ge \frac{3}{2}\) after correcting for the sign flip when dividing by a negative number. The final solution confirms that \(x\) must be greater than or equal to \(1.5\).

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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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