Week 25: Two-statement problems, spheres, and equal-distance motion

Field Book

90 to 92

Due

Test 22 due

Method

Paper first, every step

Word problems get a second sentence this week, and that second sentence is not decoration. Each sentence becomes one equation, and two equations with two unknowns is something you already know how to solve. There is also one rule here that students lose points on for the rest of their lives, so learn it now: multiplying or dividing an inequality by a negative flips the sign.

What you have to be able to do

Turn a two-sentence problem into two equations
Flip the inequality sign when multiplying or dividing by a negative
Solve a motion problem where two distances are equal

The procedures

Two statements of equality

Each sentence in the problem gives you one equation. Read them one at a time.
Name your unknowns first, in writing, before you start.
Two equations, two unknowns. Solve by substitution or elimination, whichever is cleaner.

Inequalities and negatives

Adding or subtracting on both sides never changes the direction of an inequality.
Multiplying or dividing by a positive never changes it either.
Multiplying or dividing by a negative flips it. Every time, no exceptions, and it is the single most common lost point in this chapter.

Equal-distance motion

Rate times time equals distance. Write that across the top of your work.
Build a row for each traveler with their rate and their time.
If the two distances are equal, set the two expressions equal and solve for the time.

Worked

WORKED

Solve −3x > 12.
Divide both sides by −3. Because the divisor is negative, the sign flips.
ANSWER: x < −4

WORKED

A car leaves at 40 mph. Another leaves an hour later at 60 mph on the same road. When does the second catch the first?
Distances are equal when it catches up: 40t = 60(t − 1).
40t = 60t − 60, so −20t = −60.
ANSWER: t = 3 hours after the first car left

Terms

Term	Meaning
uniform motion	travel at a steady rate, where rate times time equals distance
sphere	a solid ball, volume four thirds pi r cubed
inequality	a statement that one quantity is larger or smaller than another

Check yourself

1. Solve −2x < 10

2. Solve 5x − 3 > 12

3. Two runners at 6 and 8 mph, one starting an hour ahead. What equation sets their distances equal?

reveal answers

x > −5 · x > 3 · 6t = 8(t − 1)

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.