In woodshop class, cut pieces of wood to within 3/16 inch of the specified measurements. Let (s - x) represent the difference between the specification s and the actual measure x of a cut piece. (a) Write an absolute value inequality to describe the values of x that meet the specifications. (b) The specified length of one piece of wood is s = 5 1/8 inches. State the range of acceptable lengths for this piece.

(a) The absolute value inequality to describe the values of x that meet the specifications, where the piece needs to be within 3/16 inch of the measurement s, is:

|s - x| ≤ 3/16

This inequality states that the deviation from the specification, s - x, can be no more than 3/16 inch, either above or below the specification.

(b) Given the specified length s = 5 1/8 inches (or 5.125 inches when converted to decimal), we can find the range of acceptable lengths by applying the inequality from part (a).

First, convert 5 1/8 inches to an improper fraction or a decimal for easier calculation: 5 1/8 = 5 + 1/8 = 40/8 + 1/8 = 41/8 inches 41/8 inches is the center point of the range.

Now use the absolute value inequality |s - x| ≤ 3/16 with s = 41/8 to find the acceptable range:

-3/16 ≤ (41/8 - x) ≤ 3/16

To find the lower end of the range, solve for x using the left inequality: 41/8 - x ≥ -3/16 (multiply both sides by -1 and flip inequality) x ≤ 41/8 + 3/16 (note that we need to find a common denominator, which is 16) x ≤ (41*2)/(8*2) + 3/16 x ≤ 82/16 + 3/16 x ≤ 85/16

Thus, the lower bound for x is 85/16 inches.

So the range of acceptable lengths for this piece is from 79/16 inches to 85/16 inches.

Extra: Understanding absolute value and inequalities is a fundamental concept in algebra. The absolute value of a number is its distance from zero on the number line, regardless of direction. The inequality is a mathematical statement that one value is less than, greater than, less than or equal to, or greater than or equal to another value.

When using absolute value inequalities to describe tolerances in measurements, such as in a woodshop class, it allows us to encapsulate both the upper and lower bounds of what is acceptable. In practical applications like building and manufacturing, maintaining parts within certain tolerances is critical for ensuring proper fit and function. By using absolute values, we can easily express these bounds without having to write out two separate inequalities.

In the example provided, the use of an absolute value inequality simplifies the expression of the range of acceptable values that are slightly smaller or larger than the specified measurement. It's a handy tool that has real-world applications, especially in fields that rely on precision such as engineering, construction, and fabrication.