A binary operation \ast on the set of real numbers \mathbb{R} \setminus \{2\} is defined b… — JAMB UTME

A binary operation $\ast$ on the set of real numbers $\mathbb{R} \setminus \{2\}$ is defined by $a \ast b = 2a + 2b - ab - 2$ . If $y^{-1}$ represents the inverse of an element $y$ under the operation $\ast$ , find the value of $x$ such that $(x \ast 3) \ast 4^{-1} = 5$ .

Answer: 8

Answer

The value of x is 8.

To solve for x, first find the identity element by solving a * e = a, yielding e = 1. Next, compute 4^{-1} from 4 * 4^{-1} = 1, which gives 4^{-1} = 2.5. Then simplify x * 3 to 4 - x. Finally, substitute into (4 - x) * 2.5 = 5 and solve for x to get 8.

Step-by-Step Solution

Find the identity element e of the operation

e = 1

By definition, a * e = a. Substituting into the operation gives 2a + 2e - ae - 2 = a, which simplifies to (a - 2)(1 - e) = 0. Since a != 2, e must equal 1.

Calculate the inverse element 4^{-1}

4^{-1} = 2.5

By definition of an inverse element, 4 * 4^{-1} = e = 1. Applying the operation formula yields 2(4) + 2(4^{-1}) - 4(4^{-1}) - 2 = 1, which simplifies to 6 - 2(4^{-1}) = 1, so 4^{-1} = 2.5.

Express x * 3 in terms of x

x * 3 = 4 - x

Evaluating x * 3 using the operational definition gives 2x + 2(3) - 3x - 2 = 4 - x.

Solve the main equation (x * 3) * 4^{-1} = 5 for x

x = 8

Substituting x * 3 = 4 - x and 4^{-1} = 2.5 into the equation yields (4 - x) * 2.5 = 5. Applying the operation gives 2(4 - x) + 2(2.5) - 2.5(4 - x) - 2 = 5, which simplifies to 1 + 0.5x = 5, giving x = 8.

Key Concept

Identity and Inverse Elements in Binary Operations

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