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

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

Step 1: Rewrite the Absolute Value Inequality

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

Step 2: Solve for $T$

Now, add 52 to each part of the inequality to isolate $T$: $-28 + 52 \leq T \leq 28 + 52$ $24 \leq T \leq 80$ This means the monthly average temperature $T$ 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 $|T - 52| \leq 28$, we can conclude:

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

Summary

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

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