When solving compound inequalities, you do not need to make \( x \) positive if it is negative. Instead, you should isolate \( x \) based on the operations you are performing, keeping its sign according to the steps of your solution.
In a compound inequality, you generally follow these steps:
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Isolate the variable: Perform the same operations on all parts of the inequality to isolate \( x \). This may involve adding, subtracting, multiplying, or dividing by numbers.
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Pay attention to signs when multiplying/dividing: If you multiply or divide by a negative number, you must flip the inequality sign.
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Keep the sign of \( x \): If \( x \) ends up negative after the operations, you leave it as negative. The solution should reflect the actual values that satisfy the inequality.
Example: Given the compound inequality \(-3 < 2x + 1 < 5\):
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First, isolate \( 2x \): \[ -4 < 2x < 4 \]
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Next, divide each part by 2 to solve for \( x \): \[ -2 < x < 2 \]
In this case, \( x \) can be negative (i.e., it is between -2 and 2), and you leave it as is.
So, in summary, you do not change the sign of \( x \) to make it positive; you solve for \( x \) according to the operations performed.
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