Week 26: Rational expressions, motion that adds up, and curves

Field Book

93 to 96

Due

Test 23 due

Method

Paper first, every step

Rational expressions are fractions with algebra in them, and they follow the exact fraction rules you learned years ago. Factor first, cancel matching factors, then multiply. The motion problems change shape this week: instead of two distances being equal, they add up to a total, which is what happens when two people travel toward each other.

What you have to be able to do

Multiply and divide rational expressions
Solve a motion problem where two distances add to a total
Recognize the shape of a non-linear graph

The procedures

Multiplying and dividing rational expressions

Factor every numerator and every denominator completely before you do anything else.
Cancel any factor that appears both on top and on the bottom. Factors cancel, terms do not.
Multiply what remains. To divide, flip the second fraction over first and then multiply.

Motion where distances add

Rate times time equals distance, one row per traveler.
When two travelers move toward each other, their distances add up to the gap between them.
Write distance plus distance equals total, then solve for the time.

Shapes of non-linear graphs

A squared term bends the graph into a parabola, a U shape.
A variable in the denominator gives a hyperbola, two separate curves.
Absolute value gives a sharp V. Learning the shapes lets you check a graph at a glance.

Worked

WORKED

Multiply (x3)(6x²).
Multiply straight across: 6x over 3x².
Cancel a 3 and an x from top and bottom.
ANSWER: 2x

WORKED

Two cars 360 miles apart drive toward each other at 50 and 70 mph. When do they meet?
Distances add to the gap: 50t + 70t = 360.
120t = 360.
ANSWER: t = 3 hours

Terms

Term	Meaning
rational expression	a fraction whose numerator and denominator are polynomials
parabola	the U-shaped curve a squared term produces
hyperbola	the two-branch curve a variable in the denominator produces

Check yourself

1. Multiply (x2)(4x)

2. Two cyclists 60 miles apart ride toward each other at 12 and 18 mph. When do they meet?

3. What shape does y = x² + 1 make?

reveal answers

2 · 2 hours · a parabola

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.