During a winter storm, nearly a foot (not the mean) of snowfall covered parts of central Indiana. While some areas received as little as 5 ¼ inches, Indiana Online recorded the most, 17 ¾ inches at the Pyramids.
Part A: Write an absolute value inequality that describes the range of snowfall amounts in various locations. Use x to represent measurable snowfall in inches.
Part B: Solve the absolute value inequality for x. Express your answer as a compound inequality.
Part A: The absolute value inequality that describes the range of snowfall amounts in various locations is |x - 12.375| ≤ 5.375.
Part B: To solve the absolute value inequality, we can split it into two separate inequalities:
x - 12.375 ≤ 5.375 (1)
and
-(x - 12.375) ≤ 5.375 (2)
Solving inequality (1):
x - 12.375 ≤ 5.375
x ≤ 5.375 + 12.375
x ≤ 17.75
Solving inequality (2):
-(x - 12.375) ≤ 5.375
-x + 12.375 ≤ 5.375
-x ≤ 5.375 - 12.375
-x ≤ -7
Multiplying both sides by -1 (which reverses the inequality):
x ≥ 7
Therefore, the solution to the absolute value inequality is 7 ≤ x ≤ 17.75, expressed as a compound inequality.
Part A: The absolute value inequality that describes the range of snowfall amounts in various locations can be written as:
| x - 17 ¾ | ≤ 12
Part B: To solve the absolute value inequality, we need to consider two cases:
Case 1: x - 17 ¾ ≥ 0
In this case, the absolute value becomes x - 17 ¾ ≤ 12. Solving for x, we have:
x ≤ 17 ¾ + 12
x ≤ 29 ¾
Case 2: -(x - 17 ¾) ≥ 0
In this case, the absolute value becomes -x + 17 ¾ ≤ 12. Solving for x, we have:
x ≥ 17 ¾ - 12
x ≥ 5 ¾
Therefore, the solution to the absolute value inequality is:
5 ¾ ≤ x ≤ 29 ¾