Question

1. for 90% of the last 30 years, the rainfall at shell beach has varied no more than 6.5 inches from its mean value of 24 inches. write and solve an absolute value inequality to describe the rainfall in the other 10% of the last 30 years, and graph the solution on a number line.

Explanation:

Step1: Define the variable and inequality

Let \( x \) be the rainfall in inches. For 90% of the years, \( |x - 24| \leq 6.5 \). For the other 10%, the inequality is the complement: \( |x - 24| > 6.5 \).

Step2: Solve the absolute - value inequality

The absolute - value inequality \( |x - 24|>6.5 \) is equivalent to two separate inequalities:

• Case 1: \( x - 24>6.5 \)

Add 24 to both sides of the inequality: \( x-24 + 24>6.5+24 \), so \( x > 30.5 \).

• Case 2: \( x - 24<-6.5 \)

Add 24 to both sides of the inequality: \( x-24 + 24<-6.5 + 24 \), so \( x<17.5 \).

Answer:

The absolute - value inequality is \( |x - 24|>6.5 \), and its solution is \( x < 17.5 \) or \( x>30.5 \). To graph the solution on a number line:

• Draw an open circle at \( 17.5 \) and shade the region to the left of \( 17.5 \).
• Draw an open circle at \( 30.5 \) and shade the region to the right of \( 30.5 \).