The temperature, \(t,\) in degrees Fahrenheit, in a certain town on a certain spring day satisfies the inequality \(|t-24| \leq 30 .\) Which of the following temperatures, in degrees Fahrenheit, is NOT in this range? F. \(-10\) G. \(-10\) H. \(-5\) J. 0 K. 54

Understanding absolute value inequalities is crucial for various mathematical concepts, and it's particularly important in solving temperature range problems. An absolute value describes the distance of a number to zero on a number line, without considering direction. In essence, it's always non-negative. For instance, the absolute value of both \( -5 \) and \( 5 \) is \( 5 \).

When dealing with inequalities that include an absolute value, like \( |t - 24| \), we're being asked to find all values of \( t \) that, after subtracting 24 and taking the absolute value, result in a number less than or equal to 30. Conceptually, you can think of it as finding a segment on the number line that is centered around 24 and spans 30 units in both directions. This approach works with any absolute value inequality: first, split the equation into two scenarios (one for the positive case and one for the negative case) and solve each one to find your solution set.

Why Split It?

• Splitting the inequality helps us accommodate the definition of absolute value, which is essentially an if-else statement in itself: if the expression inside is non-negative, leave it as is; if it's negative, make it positive.
• This gives us two separate inequalities to solve, and the solution to the original absolute value inequality is the intersection of the solutions to these two separate inequalities.
• By doing this, we ensure that we find all possible solution values that are within a certain distance from the 'center' point, in this case, 24.

The temperature range problem provided offers a real-life context for applying absolute value inequalities. These types of problems often involve finding a range of acceptable values within certain bounds, mirroring real-world situations like determining acceptable temperature levels, noise levels, or even financial budgets.

In our example, the temperature in a specific location on a spring day is given by the inequality \( |t - 24| \leq 30 \), representing that the actual temperature can deviate up to 30 degrees Fahrenheit from a base of 24 degrees. By solving the absolute value inequality, we determine that the acceptable temperature ranges from \( -6 \) to \( 54 \) degrees Fahrenheit. These problems, therefore, require us to apply our knowledge of absolute values to create a practical result: delineating the range of an acceptable environment base on the given conditions.

Real-World Applications

• Temperature control systems in buildings often have to maintain temperatures within a specific range for comfort or equipment operation standards.
• In manufacturing, materials often require storage at temperatures within a certain safe range to maintain integrity.

Solving inequalities, just like equations, requires a step-by-step approach to isolate the variable of interest. When the inequality involves an absolute value, additional steps are needed to account for the nature of absolute values. Here's a simplified version of the steps used in the exercise to solve the absolute value inequality:

1. Understand the structure of the inequality and identify the absolute value component.
2. Split the inequality into two separate cases based on the nature of the absolute value: one for the positive outcome and one for the negative outcome.
3. Solve each case just like any other linear inequality.
4. Combine the solutions from both cases to create a solution set that satisfies the original inequality.
5. Review the range to ensure it matches any real-world constraints in the problem.

These steps are a template for handling absolute value inequalities, which you can apply to a vast array of problems. It's important to work through each step methodically to avoid errors and to make sure you're considering all possible solutions within the defined constraints.