THE COMPOUND // FIELD NOTES // WEEK 26
Joint variation and absolute value inequalities
Field Book
Lessons 96 to 99
Due
Test 23 due ยท Supplement 26
Method
Paper first, every step
Vocab 3 runs in the Yard, the last scheduled walk. Absolute value measures distance from zero: read every inequality as a distance sentence.
What you have to be able to do
- Set up joint and combined variation
- Find specific numbers from stated relationships
- Solve absolute value inequalities, including the empty and all-real cases
The procedures
Joint and combined variationField Book Lesson 96
- Varies jointly means varies directly with both: y = kxz.
- Combined variation stacks direct on top and inverse below. Irrational roots still pair up.
Elimination when dictatedField Book Lesson 97
- Some systems are built for elimination: aligned coefficients say so.
- Do not force substitution where elimination is two lines shorter.
Finding numbersField Book Lesson 98
- Name the number N, translate each clause, and solve.
- Check the answer against the original sentence, not your equation.
Absolute value inequalitiesField Book Lesson 99
- |x| < D means within D of zero: the segment between -D and D.
- |x| > D means beyond D: two rays. Against a negative bound, the answer is everything or nothing: decide by reading.
Worked: within three
- Solve |x| < 3.
- Distance from zero under 3: -3 < x < 3.
Terms on the record
| Term | Meaning |
|---|
| joint variation | varying directly with two quantities at once |
| absolute value | the distance from a number to zero |
| empty set | the solution set with nothing in it |