Given x = (1)/(2) and 180^ < x < 270^. This means x is in the third quadrant, where both x and x are negative.

Alright jaay — let's do this. Step 1: Determine x and x. Given x = (1)/(2) and 180^ < x < 270^. This means x is in the third quadrant, where both x and x are negative. We can form a right-angled triangle where the opposite side is 1 and the adjacent side is 2. The hypotenuse h = sqrt(1^2 + 2^2) = sqrt(1+4) = sqrt(5). Therefore, in the third quadrant: x = -(1)/(sqrt(5)) x = -(2)/(sqrt(5)) Step 2: Calculate 2x. Using the double angle formula 2x = 2 x x: 2x = 2 (-(1)/(sqrt(5))) (-(2)/(sqrt(5))) = 2 ((2)/(5)) = (4)/(5) Step 3: Substitute the values into the expression. The expression to evaluate is (3 2x + x)/( 2x - 2 x). Substitute the values of 2x and x: (3 (4)/(5)) + (-(2)/(sqrt(5)))((4)/(5)) - 2 (-(2)/(sqrt(5))) = (12)/(5) - (2)/(sqrt(5))(4)/(5) + (4)/(sqrt(5)) Step 4: Simplify the complex fraction. To combine the terms in the numerator and denominator, we rationalize the sqrt(5) terms: (2)/(sqrt(5)) = 2sqrt(5)5 and (4)/(sqrt(5)) = 4sqrt(5)5 Substitute these back: (12)/(5) - 2sqrt(5)5(4)/(5) + 4sqrt(5)5 = 12 - 2sqrt(5)54 + 4sqrt(5)5 = 12 - 2sqrt(5)4 + 4sqrt(5) Step 5: Rationalize the denominator. Factor out common terms and then multiply the numerator and denominator by the conjugate of the denominator, which is (1 - sqrt(5)): 2(6 - sqrt(5))4(1 + sqrt(5)) = 6 - sqrt(5)2(1 + sqrt(5)) 6 - sqrt(5)2(1 + sqrt(5)) × 1 - sqrt(5)1 - sqrt(5) Numerator: (6 - sqrt(5))(1 - sqrt(5)) = 6 - 6sqrt(5) - sqrt(5) + (sqrt(5))^2 = 6 - 7sqrt(5) + 5 = 11 - 7sqrt(5) Denominator: 2(1 + sqrt(5))(1 - sqrt(5)) = 2(1^2 - (sqrt(5))^2) = 2(1 - 5) = 2(-4) = -8 Step 6: Write the final expression in surd form. 11 - 7sqrt(5)-8 = -11 - 7sqrt(5)8 = -(11 - 7sqrt(5))8 = -11 + 7sqrt(5)8 = 7sqrt(5) - 118 Send me the next one 📸