Solving the Inequality: How to Find the Solution for (a-x+1)(a-x+2) ≤ a?

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Discussion Overview

The discussion revolves around solving the inequality (a-x+1)(a-x+2) ≤ a, where a is a constant with an unknown value. Participants explore methods for manipulating the inequality and expressing x in terms of a.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant asks how to solve the inequality, indicating a need for assistance.
  • Another participant suggests expanding the left-hand side and rearranging the inequality in terms of x, while noting important rules regarding inequalities, such as the implications of dividing by zero or negative numbers.
  • A third participant proposes a method involving the difference of two squares to express the inequality in a different form, specifically suggesting the transformation to ((a - x + 3/2) - 1/2)((a - x + 3/2) + 1/2) ≤ a.
  • The same third participant reiterates their approach, indicating a sense of affirmation in their method.

Areas of Agreement / Disagreement

There is no consensus on a single method for solving the inequality, as multiple approaches are presented without resolution of which is preferable.

Contextual Notes

Participants have not fully explored the implications of the constant a, nor have they resolved any potential issues related to the conditions under which the inequality holds.

Who May Find This Useful

Individuals interested in mathematical inequalities, algebraic manipulation, or those seeking assistance with similar problems may find this discussion beneficial.

nightking
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How can I solve this inequality?

(a-x+1)(a-x+2) ≤ a

where a is a constant with unknown value.

Thanks in advance.
 
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Hey nightking and welcome to the forums.

You need to expand out the left hand side and then put on side completely in terms of x.

The rules for inequalities are that you can't divide any side by zero (you also have to make sure any variables you have are not zero either if you want to divide), if you divide by a negative number you flip the inequality sign, if you subtract or add a term the sign doesn't change.
 
If you want to find x in terms of a, I would start with
((a - x + 3/2) - 1/2)((a - x + 3/2) + 1/2) ≤ a

The left hand side is then the difference of two squares...
 
AlephZero said:
If you want to find x in terms of a, I would start with
((a - x + 3/2) - 1/2)((a - x + 3/2) + 1/2) ≤ a

The left hand side is then the difference of two squares...

Brilliant. Thanks!
 

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