Depth · one concept from first principles · step 2 of 4
Fix the signs, then compare reciprocals
Three nonzero rational numbers , , satisfy , , , and . List the following numbers from greatest to least.
Number system
Your answer
How to solve
Strategy Identify Subproblems — Split the job in two. First decide the sign of each of x, y, z, ruling out the impossible sign patterns. Then sort the five numbers into positives, zero, and negatives, and compare only within each group, where a comparison of sizes is enough.
1STEP 1
z is the largest
x-z ＜ 0 means x ＜ z, and z-y ＞ 0 means y ＜ z.
Whatever the signs are, z is bigger than both x and y.
6.NS.C.7Identify Subproblems2STEP 2
Decide the signs
A positive product has zero or two negatives. All positive breaks x+y ＜ 0, so two are negative, and the largest, z, is the positive one.
z ＞ 0, x ＜ 0, y ＜ 0
Two negatives and one positive is the only pattern that fits every condition.
7.NS.A.2Eliminate Possibilitiesis positive while and are both negative.
Why?
Each negative factor flips the sign of a product and two flips land back on positive, so a product of three numbers is positive only when it has no negative factors or exactly two.
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
Why?
All three cannot be positive, because then would be more than , which is more than , contradicting .
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
The one positive number must be , because is the largest of the three.
Why?
means and means , since sliding both sides by the same amount keeps their order.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
A positive number is above and a negative one below it, so the single positive number is above both others and is the largest.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
3STEP 3
Sort the five numbers by sign
Use x ＜ 0, x+y ＜ 0, z ＞ 0, and x-z ＜ 0.
-1/x ＞ 0, -1/(x+y) ＞ 0, -1/z ＜ 0, 1/(x-z) ＜ 0
Two positives, then 0, then two negatives.
7.NS.A.2Identify Subproblems4STEP 4
Compare the two positives
Since y ＜ 0, adding it moves x further left: x+y ＜ x ＜ 0. So x+y has the larger size, and 1 over a larger size is smaller.
0 ＜ -x ＜ -(x+y) → -1/(x+y) ＜ -1/x
-1/x is the greatest of the five.
6.NS.C.7Identify Subproblems5STEP 5
Compare the two negatives
Since x ＜ 0, we have z-x ＞ z ＞ 0. Note 1/(x-z) = -1/(z-x), and 1/(z-x) ＜ 1/z.
0 ＜ z ＜ z-x → -1/z ＜ -1/(z-x)=1/(x-z)
-1/z is the least of the five.
6.NS.C.7Identify SubproblemsAnswer
-1/x, -1/(x+y), 0, 1/(x-z), -1/z
Test values that satisfy every condition: x=-1, y=-2, z=1 (then x+y=-3, x-z=-2, z-y=3, xyz=2). The five numbers become 1, 1/3, -1, -1/2, 0, and from greatest to least they read 1, 1/3, 0, -1/2, -1, matching the order found.
Takeaway
Signs first, sizes second: group the numbers by sign and only compare inside each group.
- z is the largest
- Decide the signs
- Sort the five numbers by sign
- Compare the two positives
- Compare the two negatives
Where next?
Level upsuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.