Depth · one concept from first principles · step 2 of 4

Peel a nested defined operation from the outside

For two numbers $a$ and $b$ , define $a \circledcirc b=a+b-ab$ . Find the value of $x$ that satisfies $\{(x-3)\circledcirc 7\}\circledcirc 2=1$ .

Expressions

Your answer

How to solve

Strategy Work Backwards — Give the inner block a name, P. Undo the outer operation first to learn what P must be, then undo the inner operation to get x. Each layer is a short linear equation.

1STEP 1

Name the inside

Let P=(x-3)⊚ 7, so the equation reads P⊚ 2=1.

P⊚ 2=1

2STEP 2

Undo the outer layer

By the definition, P⊚ 2=P+2-2P=2-P. Set it equal to 1.

2-P=1 → P=1

3STEP 3

Write the inner layer

Use the definition with a=x-3 and b=7.

(x-3)⊚ 7=(x-3)+7-7(x-3)=7-6(x-3)

4STEP 4

Undo the inner layer

Set it equal to 1 and solve.

7-6(x-3)=1 → 6(x-3)=6 → x-3=1 → x=4

Answer

At x=4: 1⊚ 7=1+7-7=1, then 1⊚ 2=1+2-2=1. The result is 1, as required.

Takeaway

For an operation inside an operation, work from the outside in: find what the inside has to be, then solve for x.

Name the inside
Undo the outer layer
Write the inner layer
Undo the inner layer

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