Question

№1 Решите неравенство: а) 9x-11 > 5(2x-3) б) x^2+7x-8>0 №2 Решите уравнение: а) 3x-2√x-8x=0 _____ б) 2x+15=х _____ №3 Найдите область определения выражения (1)/√4x+3 №4 Докажите, что функция y= (3x-5)/2 возрастает. №5 При каких значениях параметра P уравнение px^2-2px+9=0 имеет два корня?

Answer 1

a)9x-11 >5(2x-3) 

 x<4

 

 6)x^2-7x-8 >0

D=49-4*1*-8=49+32=V81=9

 x=(7+9)/2=8

x2= (7-9)/2= -1 

 

 

 (-oo ; -1)  U (8; +oo)

 

 

 

 

 

 

a)3x-2Vx-8x=0

    -5x-2Vx=0

   -2Vx=5x

    4x=25x^2

 

    4=25x

   x=4/25

 

 

 

 

 

 6)2x+15=x 

    x=-15

 

 

 

 

 N3) 1/V4x+3

 V4x+3>0

 4x+3>0

 4x>-3 

 x>-3/4

 

 

 

 

Функция возрастает  потому что

y=kx+b

это уравнение прямой  а коэффициент k>0

 

 

 

то есть 3/2 >0 

 

 

 

 

 

 px^2-2px+9=0

D=4p^2-36p=0

p^2-9p=0

p(p-9)=0

p=0

p=9

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer 2

№1: a) 9x - 11 > 5(2x - 3) Simplifying the inequality: 9x - 11 > 10x - 15 Subtracting 9x from both sides: -11 > x - 15 Adding 15 to both sides: 4 > x So the solution is x < 4.

b) x^2 + 7x - 8 > 0 We can factor the left-hand side: (x + 8)(x - 1) > 0 The inequality is true when both factors have the same sign, either both positive or both negative.

When x > 1, both factors are positive, so the inequality is true. When x < -8, both factors are negative, so the inequality is also true. When -8 < x < 1, the left-hand side is negative, so the inequality is false.

Therefore, the solution is x < -8 or x > 1.

№2: a) 3x - 2√x - 8x = 0 Simplifying the left-hand side: -2√x - 5x = 0 Dividing both sides by -5: √x = -2x/5 Squaring both sides: x = 4x^2/25 Multiplying both sides by 25: 25x = 4x^2 Dividing both sides by x: 25 = 4x x = 25/4 However, we need to check that this solution satisfies the original equation since we squared both sides of an equation: 3(25/4) - 2√(25/4) - 8(25/4) = 0 75/4 - 5 - 50 = 0 So the solution is x = 25/4.

b) 2x + 15 = х Substituting x + 15 for х: 2x + 15 = x + 15 Subtracting x and 15 from both sides: x = 0 So the solution is x = 0.

№3: The expression (1)/√(4x + 3) is defined only when the radicand is non-negative, so we need to solve the inequality: 4x + 3 ≥ 0 Subtracting 3 from both sides: 4x ≥ -3 Dividing both sides by 4 (since 4 is positive): x ≥ -3/4 Therefore, the domain of the expression is x ≥ -3/4.

№4: To show that the function y = (3x - 5)/2 is increasing, we need to show that for any two values of x, say x1 and x2, such that x1 < x2, the corresponding values of y, y1 and y2, satisfy y1 < y2.

Let's consider the difference between y2 and y1: y2 - y1 = [(3x2 - 5)/2] - [(3x1 - 5)/2] Simplifying: y2 - y1 = (3x2 - 5 - 3x1 + 5)/2 y2 - y1 = 3(x2 - x1)/2 y2 - y1 = (3/2)(x2 - x1)

Since x2 - x1 > 0 (since x2 > x1), then y2 - y1 > 0, which means y2 > y1. Therefore, the function is