Question

In Exercises \(83-85,\) choose the statement that is true about the given numbers. (A) The number in column A is greater. (B) The number in column B is greater. (C) The two numbers are equal. (D) The relationship cannot be determined from the given information. Column A Column B $$ |x|+8 | x+8 $$

Short answer

(D) The relationship cannot be determined from the given information.

Step-by-step solution

Break down the problem into two cases

The problem must be separated into two distinct cases because the sign of x is essential in determining the relationship. Two cases need to be considered: when \(x \geq -8\) and \(x< -8\).

Examine case when x is greater than or equal to -8

If \(x\geq -8\), then \(|x|+8\) equals \(x+8\). Because \(|x|\) equals to \(x\) itself, once \(x\geq -8\). As a result, the number in Column A equals the number in Column B for this case.

Examine case when x is less than -8

If \(x< -8\), \(|x|+8\) equals \(-x+8\). This occurs because \(|x|\) equals \(-x\) if \(x<0\). However, the number in Column B, \(x + 8\), is less than the number in Column A, \(-x+8\), for \(x< -8\). As a result, the number in Column A is greater for this case.

Bring together the results of each case

The number in Column A will be greater when \(x< -8\), while the two numbers will be equal when \(x\geq -8\). As a result, it's impossible to decide from the given information whether the number in Column A or Column B is greater; it depends on the actual value of x.