Depth · one concept from first principles · step 2 of 4
ac=1 makes a and c same-sign reciprocals
Three rational numbers , , satisfy , and . If , how are , and ordered? Write the order using inequality signs.
Number system
Your answer
How to solve
Strategy Identify Subproblems — Work out signs before sizes. Because ac = 1, the product abc is just b, which settles the sign of b. The sum then forces a and c to be positive, and the reciprocal relation turns |a| ＜ 1 into c ＞ 1.
1STEP 1
a and c share a sign
Their product 1 is positive, so they are both positive or both negative.
ac = 1 ＞ 0
Only like signs give a positive product.
7.NS.A.2Identify Subproblems2STEP 2
The sign of b
Group the product as b × (ac); since ac = 1, the whole product equals b.
a × b × c = b × (ac) = b × 1 = b ＜ 0
abc ＜ 0 means simply b ＜ 0.
7.NS.A.2Convert To AlgebraThe product is just , so .
Why?
A product can be reordered and regrouped freely, so .
Why?
Swapping and does not change the product.
🧱Commutativity of multiplicationa rows of b and b rows of a count the same array of dots — rows and columns just swap.
Why?
Multiplying and first and then gives the same product as any other grouping.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
Why?
Since , the product is , and multiplying by one changes nothing.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
3STEP 3
The signs of a and c
From a + b + c ＞ 0 we get a + c ＞ -b, and -b is positive. So a + c ＞ 0, which is impossible if a and c are both negative.
a + c ＞ -b ＞ 0 → a ＞ 0, c ＞ 0
So a ＞ 0 and c ＞ 0.
6.NS.C.7Identify Subproblems4STEP 4
Compare a and c
c is the reciprocal of a. With 0 ＜ a ＜ 1, dividing 1 by a number less than 1 gives more than 1.
c = 1/a, 0 ＜ a ＜ 1 → c ＞ 1
For example a = 1/2 gives c = 2.
7.NS.A.2Convert To Algebra5STEP 5
Put them in order
b is negative, a is between 0 and 1, and c is greater than 1.
b ＜ 0 ＜ a ＜ 1 ＜ c
So b ＜ a ＜ c.
6.NS.C.7Identify SubproblemsAnswer
b ＜ a ＜ c
Try a = 1/2, c = 2, b = -1: ac = 1, abc = -1 ＜ 0, a + b + c = 3/2 ＞ 0, |a| ＜ 1. The order is -1 ＜ 1/2 ＜ 2, that is b ＜ a ＜ c.
Takeaway
When two numbers multiply to 1, they share a sign and one is big exactly when the other is small.
- a and c share a sign
- The sign of b
- The signs of a and c
- Compare a and c
- Put them in order
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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