The ideal mass for a piece of chocolate is 2.5 ounces. The actual mass for a production can vary by, at most, 0.08 ounces. Create an absolute value inequality and solve it to determine the range of chocolate masses. I don't understand this
|x-a| is the distance from x to a, either plus or minus
Now, consider that
|x| = x if x >= 0
|x| = -x if x < 0
and you have
|m - 2.5| <= 0.08
so, if m > 2.5,
m-2.5 <= 0.08
m <= 2.58
Otherwise,
-(m-2.5) <= 0.08
m-2.5 >= 0.08
m >= 2.42
We can write this as
2.5-0.08 <= m <= 2.5+0.08
2.42 <= m <= 2.58
To solve the problem, we can define an absolute value inequality to represent the range of masses. Let's call the actual mass of the chocolate "x."
The absolute value of the difference between the actual mass (x) and the ideal mass (2.5 ounces) should be less than or equal to the maximum allowed variation (+/- 0.08 ounces). This can be represented as:
| x - 2.5 | ≤ 0.08
Now we can solve this inequality.
Step 1: Set up two separate inequalities.
1) x - 2.5 ≤ 0.08
2) -(x - 2.5) ≤ 0.08
Step 2: Solve each inequality.
Inequality 1: x - 2.5 ≤ 0.08
Add 2.5 to both sides: x ≤ 2.5 + 0.08
Simplify: x ≤ 2.58
Inequality 2: -(x - 2.5) ≤ 0.08
Distribute the negative sign: -x + 2.5 ≤ 0.08
Subtract 2.5 from both sides: -x ≤ 0.08 - 2.5
Simplify: -x ≤ -2.42
Multiply both sides by -1 (this flips the inequality sign): x ≥ 2.42
Step 3: Combine the solutions.
The range of chocolate masses can be described using the values of x that satisfy both inequalities, so we take the intersection of the solutions:
2.42 ≤ x ≤ 2.58
Therefore, the range of chocolate masses is from 2.42 ounces to 2.58 ounces. Any mass within this range is acceptable.