A binary operation \Delta is defined on the set of real numbers \mathbb{R} by a \Delta b =… — JAMB UTME

A binary operation $\Delta$ is defined on the set of real numbers $\mathbb{R}$ by $a \Delta b = a + 3b - 2ab$ . If $(3 \Delta x) \Delta 1 = 7$ , what is the value of $x$ ?

$\frac{7}{3}$ Answer
B
$\frac{3}{5}$
C
$-\frac{7}{9}$
D
$-\frac{4}{3}$

Answer

\frac{7}{3}

Evaluating

3 \Delta x

gives

3 - 3x

. Then operating

(3 - 3x) \Delta 1

yields

(3 - 3x) + 3 - 2(3 - 3x) = 3x

. Equating

3x = 7

gives

x = \frac{7}{3}

Step-by-Step Solution

Evaluate the inner expression

3 \Delta x

using the definition

a \Delta b = a + 3b - 2ab

3 \Delta x = 3 + 3x - 2(3)(x) = 3 + 3x - 6x = 3 - 3x

Substitute

a = 3

and

b = x

into the operation rule.

Substitute the result

(3 - 3x)

as the first operand in the outer expression

(3 - 3x) \Delta 1

(3 - 3x) \Delta 1 = (3 - 3x) + 3(1) - 2(3 - 3x)(1)

Apply the binary operation definition with

a = 3 - 3x

and

b = 1

Expand and simplify the algebraic expression.

(3 - 3x) \Delta 1 = 3 - 3x + 3 - 6 + 6x = 3x

Distribute

-2

across

(3 - 3x)

to get

-6 + 6x

, then collect like terms.

Set the simplified expression equal to

7

and solve for

x

3x = 7 \implies x = \frac{7}{3}

Divide both sides by

3

to isolate

x

Key Concept

Non-commutative binary operation composition and algebraic equation solving

Question

Answer

Step-by-Step Solution

Key Concept

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