O-Level Equations and inequalities

Time for large pump = 8000/x min; time for small pump = 8000/(x-50) min. Small pump takes 35 min more: 8000/(x-50) = 8000/x + 35. Multiply through by x(x-50): 8000x = 8000(x-50) + 35x(x-50) 8000x = 8000x - 400000 + 35x² - 1750x 0 = 35x² - 1750x - 400000 Divide by 5: 7x² - 350x - 80000 = 0. (shown)

The line passes through A(0, 4) and B(2, 0). Gradient m = (0 - 4)/(2 - 0) = -4/2 = -2. The y-intercept is at A(0, 4), so c = 4.

Step 1: Get rid of the fractions. The two denominators are (2 - y) and (3y + 4), so multiply EVERY term by their product (2 - y)(3y + 4). (Note y ≠ 2 and y ≠ -4/3, since either of those would make a denominator zero. Check at the end that the answers avoid them.) First term: 3/(2 - y) × (2 - y)(3y + 4) = 3(3y + 4). Second term: 1/(3y + 4) × (2 - y)(3y + 4) = (2 - y). Right side: 5 × (2 - y)(3y + 4). So the equation becomes 3(3y + 4) - (2 - y) = 5(2 - y)(3y + 4). Step 2: Expand the left-hand side, being careful with the minus sign in front of the bracket. 3(3y + 4) = 9y + 12 -(2 - y) = -2 + y Left side = 9y + 12 - 2 + y = 10y + 10. Step 3: Expand the right-hand side. Do the two brackets first. (2 - y)(3y + 4) = 2(3y) + 2(4) - y(3y) - y(4) = 6y + 8 - 3y² - 4y = -3y² + 2y + 8. Now multiply by 5: 5(-3y² + 2y + 8) = -15y² + 10y + 40. So 10y + 10 = -15y² + 10y + 40. Step 4: Simplify. The term 10y appears on both sides, so subtract 10y from each side and the linear term disappears: 10 = -15y² + 40. Step 5: Rearrange for y². 15y² = 40 - 10 15y² = 30 y² = 2. Step 6: Square root both sides, and remember there are TWO square roots. y = ±√2 = ±1.41 (3 s.f.). Neither value equals 2 or -4/3, so both are genuine solutions. (Check with y = √2 ≈ 1.414: 3/(2 - 1.414) - 1/(3(1.414) + 4) = 5.121 - 0.121 = 5. Correct.)