Problem 81 - Beginning and Intermediate Algebra

Question

Emmanuel spent \(\$ 38\) on a birthday gift for his son. He plans on spending within \(\$ 5\) of that amount on his daughter's birthday gift. Let \(b\) represent the range of values for the amount he will spend on his daughter's gift. Write an absolute value inequality to represent the range for the amount of money Emmanuel will spend on his daughter's birthday gift, then solve the inequality and explain the meaning of the answer.

Step-by-Step Solution

Answer

The absolute value inequality to represent the range for the amount of money Emmanuel will spend on his daughter's birthday gift is \(|b - 38| \leq 5\). After solving the inequality, we find that \(33 \leq b \leq 43\). This means Emmanuel will spend between $33 and $43 (inclusive) on his daughter's birthday gift, ensuring the amount spent is within $5 of what he spent on his son's gift.

Step 1: Set up the absolute value inequality

Let \(b\) represent the amount Emmanuel will spend on his daughter's birthday gift. Since the amount he spends on his daughter's gift must be within \(5\) dollars of what he spent on his son's gift ($38), we have \[|b - 38| \leq 5\]

Step 2: Solve the absolute value inequality

To solve the absolute value inequality, we need to break it down into two cases: the positive case and the negative case. Positive Case: \[b - 38 \leq 5\] \[b \leq 43\] Negative Case: \[-(b - 38) \leq 5\] \[38 - b \leq 5\] \[-(b - 38) \leq 5\] \[b \geq 33\] Combining both cases, we get \[33 \leq b \leq 43\]

Step 3: Interpret the solution

The solution \(33 \leq b \leq 43\) means that Emmanuel will spend between \(33 and \)43 (inclusive) on his daughter's birthday gift. This range ensures that the amount he spends on his daughter's gift is within $5 of what he spent on his son's gift.