Math Problem Statement

The inequality describes the range of monthly average temperatures T in degrees Fahrenheit at a certain location. Find an equivalent expression and monthly average temperatures. |T - 52| ≤ 28.

Solution

To solve this inequality problem, let's analyze the expression step-by-step.

The inequality given is: T5228|T - 52| \leq 28 This absolute value inequality represents the range of monthly average temperatures, TT, centered around 52 degrees Fahrenheit, with a maximum deviation of 28 degrees.

Step 1: Rewrite the Absolute Value Inequality

An absolute value inequality T5228|T - 52| \leq 28 can be rewritten as: 28T5228-28 \leq T - 52 \leq 28

Step 2: Solve for TT

Now, add 52 to each part of the inequality to isolate TT: 28+52T28+52-28 + 52 \leq T \leq 28 + 52 24T8024 \leq T \leq 80 This means the monthly average temperature TT is between 24°F and 80°F.

Step 3: Answer the Fill-in-the-Blank Question

The problem then asks:

If the high and low monthly average temperatures satisfy the inequality, then the monthly averages are always within ___ degrees of ___ °F.

From the inequality T5228|T - 52| \leq 28, we can conclude:

  • The monthly averages are always within 28 degrees of 52°F.

Summary

  • Equivalent inequality: 24T8024 \leq T \leq 80
  • The monthly averages are within 28 degrees of 52°F.

Would you like a more detailed explanation or have any questions?


Here are some related questions for further exploration:

  1. What is the range of values for TT if the inequality was T6015|T - 60| \leq 15?
  2. How would the range change if we had T52<28|T - 52| < 28 instead of 28\leq 28?
  3. Can you represent the inequality 24T8024 \leq T \leq 80 on a number line?
  4. How do you solve inequalities with different constants, like T4035|T - 40| \leq 35?
  5. What is the impact on the range of TT if we increase or decrease the value of 28 in T5228|T - 52| \leq 28?

Tip: When solving absolute value inequalities, remember to split them into two inequalities without the absolute value symbol, then solve each part individually.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Absolute Value

Formulas

Absolute value inequality |T - a| ≤ b can be rewritten as -b ≤ T - a ≤ b

Theorems

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Suitable Grade Level

Grades 8-10