Suppose we want to choose a value of 
x
 within 4 units of 12. [This means a value of 
x
 that is less than 4 units away from 12.]

Think about some values of 
x
 that meet this constraint.

Write an inequality that represents all values of 
x
 that meet this constraint.

To find values of \( x \) that are within 4 units of 12, we need to set up an inequality that captures this condition.

"Within 4 units" means that the distance between \( x \) and 12 is less than or equal to 4. Mathematically, the absolute value of the difference between \( x \) and 12 must be less than or equal to 4:

\[
|x - 12| \leq 4
\]

Now, let's break down the absolute value inequality. The absolute value inequality \( |x - 12| \leq 4 \) means that \( x - 12 \) lies between -4 and 4:

\[
-4 \leq x - 12 \leq 4
\]

Next, solve for \( x \) by adding 12 to all parts of the inequality:

\[
-4 + 12 \leq x \leq 4 + 12
\]

\[
8 \leq x \leq 16
\]

Thus, the inequality representing all values of \( x \) that are within 4 units of 12 is:

\[
8 \leq x \leq 16
\]

Would you like further details or have any questions?

Here are some related questions:
1. How would the inequality change if the range were "within 5 units" of 12?
2. What values of \( x \) would satisfy \( |x - 12| > 4 \)?
3. What is the geometric interpretation of \( |x - 12| \leq 4 \)?
4. How would the inequality look for values of \( x \) within 4 units of a different number, like 20?
5. Can you graph the solution \( 8 \leq x \leq 16 \) on a number line?

**Tip:** When dealing with absolute value inequalities, think of them as describing a range or distance around a central point.

Suppose we want to choose a value of x within 4 units of 12. [This means a value of x that is less than 4 units away from 12.] Think about some values of x that meet this constraint. Write an inequality that represents all values of x that meet this constraint.

Learn how to solve the inequality |x - 12| ≤ 4, which represents values of x within 4 units of 12. This problem covers key algebraic concepts like absolute value and inequalities. Suitable for students in Grades 7-10, it provides a step-by-step breakdown and related concepts.

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