The equation $|x-8|=3$ represents the minimum and maximum percent of people in a survey who are undecided about an issue. What percent of the people in the survey are undecided about an issue?
Minimum = $\square$ %
Maximum = $\square$ %
Answer (2)
Set up two cases for the absolute value equation ∣ x − 8∣ = 3 : x − 8 = 3 and x − 8 = − 3 .
Solve the first case: x − 8 = 3 ⇒ x = 11 .
Solve the second case: x − 8 = − 3 ⇒ x = 5 .
The minimum and maximum percentages are 5% and 11%, respectively: M inim u m = 5% M a x im u m = 11% .
Explanation
Understanding the Problem We are given the equation ∣ x − 8∣ = 3 , which represents the minimum and maximum percentage of people undecided about an issue in a survey. Our goal is to find these minimum and maximum percentages.
Setting up the Cases To solve the absolute value equation ∣ x − 8∣ = 3 , we need to consider two cases:
Case 1: x − 8 = 3 Case 2: x − 8 = − 3
Solving the Equations Let's solve each case:
Case 1: x − 8 = 3 . Adding 8 to both sides, we get x = 3 + 8 = 11 .
Case 2: x − 8 = − 3 . Adding 8 to both sides, we get x = − 3 + 8 = 5 .
Determining Minimum and Maximum The two solutions for x are 11 and 5. Since these represent the minimum and maximum percentages, the minimum percentage is 5% and the maximum percentage is 11%.
Final Answer Therefore, the minimum percentage of people undecided is 5%, and the maximum percentage is 11%.
Examples
Absolute value equations are useful in various real-life scenarios. For example, in manufacturing, if a machine is set to produce parts that are 5 cm in length, and a tolerance of 0.2 cm is acceptable, the actual length x of the parts can be modeled by the equation ∣ x − 5∣ ≤ 0.2 . This ensures that the parts produced are within the acceptable range. Similarly, in finance, absolute value can be used to model deviations from a target investment return.
The minimum percentage of people undecided about the issue is 5%, and the maximum percentage is 11%. These values are derived from solving the absolute value equation ∣ x − 8∣ = 3 .
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