AMC 10 · 2017 · #2
Grade 7 algebraPick an answer.
Tool #3 (Eliminate Possibilities): five choices, only one survives, so killing four is as good as proving the fifth. Tool #14 (Extreme Principle): an expression is positive for all values exactly when its worst case (smallest possible value) is still positive — so I push each range to its low end. Tool #6 (Guess and Check): for a choice that can fail, one concrete set of numbers that makes it negative is enough to throw it out.
Read the ranges
The three ranges decide everything.
Always-positive means the worst case is still positive.
6.NS.C.7Eliminate PossibilitiesAdd the bounds for y+z
Adding the two lower bounds makes y+z always positive.
Adding the two lowest possible values gives the lowest possible sum.
7.NS.A.1Extreme PrincipleKnock out (A) and (B)
One strongly negative y breaks the first two choices.
Shrink the positive part to near zero and the negative y wins.
6.EE.B.5Guess And CheckKnock out (C) and (D) by factoring
Factoring exposes the sign of the next two.
Pull out y and read each factor's sign.
Pulling a variable out front lets each factor's sign be read on its own.
▸ Why?
A factor shared by both terms can be lifted out of the sum, leaving a simpler expression inside.
▸ Why?
A product's sign is decided by its factors' signs, so the ranges of the letters settle it at once.
Only (E) survives
Only y+z survives.
The one survivor of elimination is the answer.
6.NS.C.7Eliminate PossibilitiesAn expression is 'always positive' only if even its smallest possible value stays above zero; adding the low ends y ＞ -1 and z ＞ 1 gives y+z ＞ 0, so the answer is (E).
- Read the ranges
- Add the bounds for y+z
- Knock out (A) and (B)
- Knock out (C) and (D) by factoring
- Only (E) survives