Can you square both sides of an inequality?
Understand the Problem
The question is asking whether it is valid to square both sides of an inequality. This involves understanding the properties of inequalities and how squaring affects them, particularly if both sides are positive or negative.
Answer
It is valid to square both sides of an inequality if both sides are non-negative.
Answer for screen readers
It is valid to square both sides of an inequality if both sides are non-negative.
Steps to Solve
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Identify the Inequality Type Determine the type of inequality you have. If both sides of the inequality are positive, then you can square both sides without changing the inequality sign.
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Check the Signs If one or both sides are negative, squaring can change the direction of the inequality. For example, if you have $a < b$ and $a$ is negative while $b$ is positive, squaring both sides gives $a^2 > b^2$.
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Apply the Squaring If it is valid to square, perform the squaring operation: For example, if you have $x < y$, and both $x$ and $y$ are positive, then squaring gives $x^2 < y^2$.
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Rewrite the Inequality Use the results from squaring to rewrite the inequality to see if your conclusion still holds.
It is valid to square both sides of an inequality if both sides are non-negative.
More Information
Squaring both sides of an inequality is a common technique in algebra, but one must be cautious about the signs of the terms involved. When both terms are positive, the inequality holds. However, if one side is negative, squaring could result in an incorrect interpretation of the inequality.
Tips
- Forgetting to check the signs of both sides before squaring can lead to incorrect conclusions.
- Assuming that squaring always maintains the same direction of the inequality, which is only true for non-negative values.
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