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Explain why the solution set of the inequality |x|<5 is the intersection of two sets, while the solution set of the inequality |x|>5 is the union of two sets

User JakeCowton
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Final answer:

The inequality |x| < 5 results in an intersection of two sets (-5 < x < 5), while the inequality |x| > 5 requires a union of two sets (x < -5 or x > 5), as they represent distances from zero on the number line.

Step-by-step explanation:

The solution set of the inequality |x| < 5 is the intersection of two sets because the absolute value of x represents the distance of x from 0 on the number line, and the inequality states that this distance must be less than 5.

There are two sets of numbers that satisfy this condition: those that are less than 5 and those that are greater than -5; hence, x can range from -5 to 5. On the other hand, the solution set of the inequality |x| > 5 is the union of two sets because x must be farther from 0 than 5 units, meaning x is either greater than 5 or less than -5.

This results in two separate intervals, which must be combined through union since x can belong to either one.

User ChrHansen
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