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Chapter 7: Problem 8

Rewrite each of the statements without using a modulus sign: $$ |x|>7.5 $$

Short Answer

Expert verified
Question: Rewrite the inequality |x| > 7.5 without using the modulus sign. Answer: x < -7.5 or x > 7.5

Step by step solution

01

Case 1: x is positive

If x is positive, then we can simply remove the absolute value sign and rewrite the inequality as: $$ x > 7.5 $$
02

Case 2: x is negative

If x is negative, then the absolute value of x represents the negation of x. So, |-x| > 7.5. In this case, we have: $$ -x > 7.5 $$ Multiplying both sides by -1 (and remembering to switch the inequality sign when multiplying by a negative number) gives: $$ x < -7.5 $$
03

Combining the two cases

Now that we have the inequalities for each case, we can combine them to reflect the full range of x values that satisfy the original inequality: $$ x < -7.5 \quad \text{or} \quad x > 7.5 $$

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