Math Problem Statement
Solution
To solve this inequality problem, let's analyze the expression step-by-step.
The inequality given is: This absolute value inequality represents the range of monthly average temperatures, , centered around 52 degrees Fahrenheit, with a maximum deviation of 28 degrees.
Step 1: Rewrite the Absolute Value Inequality
An absolute value inequality can be rewritten as:
Step 2: Solve for
Now, add 52 to each part of the inequality to isolate : This means the monthly average temperature 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 , we can conclude:
- The monthly averages are always within 28 degrees of 52°F.
Summary
- Equivalent inequality:
- 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:
- What is the range of values for if the inequality was ?
- How would the range change if we had instead of ?
- Can you represent the inequality on a number line?
- How do you solve inequalities with different constants, like ?
- What is the impact on the range of if we increase or decrease the value of 28 in ?
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
-
Suitable Grade Level
Grades 8-10
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