Question

1)|x+4|=2x 2)|x-14|=8+2x 3)|x^2-4x|=3x

Answer 1

1) х+4=2х           или             -х-4=2х

-х=-4                                    -3х = 4

х=4                                       х=-4/3

Проверка:

|4+4|=8                                |-4/3+4|=-8/3

|8|=8                                    Модуль не может равняться отрицательному числу

 

х=4 

 

2) х-14=8+2х                  или               -х+14=8+2х

-х=22                                                  -3х=-6 

х=-22                                                  x = 2   

Проверка:

|-22-14|=8-44                                    |2-14|=8+4

|-36|≠-36                                          |-12| = 12

 

x=2 

 

3) x²-4x=3x                       или              -x²+4x=3x

x²-7x=0                                                x²-x = 0

x₁ = 0; x₂ = 7                                        x3= 0; x₄ = 1

Проверка:

a) |0-0|=0                                           c) 0=0

0=0                                                     d) |1-4|=3

b) |49-28|=21                                       |-3|=3

21=21                                                    3=3

 

x₁=0; x₂=1; x3 = 7 

Answer 2

1. |x+4|> или = 0  
х+4=2х
2х-х-4=0
х=4
  |x+4|<0
-(х+4)=2х
-х-4=2х
2х+х-4=0
3х=4
х=4/3 

Answer 3

Let's solve each equation step by step:

1. |x + 4| = 2x

To solve this equation, we need to consider two cases, one where the expression inside the absolute value is positive, and another where it is negative.

Case 1: x + 4 ≥ 0 In this case, the absolute value can be removed, so we have: x + 4 = 2x

Simplifying: 4 = x

Case 2: x + 4 < 0 In this case, the expression inside the absolute value becomes its negation, so we have: -(x + 4) = 2x

Simplifying: -x - 4 = 2x

Rearranging terms: 3x = -4

Dividing both sides by 3: x = -4/3

Therefore, the solutions to the equation |x + 4| = 2x are x = 4 and x = -4/3.

2. |x - 14| = 8 + 2x

Again, we consider two cases based on the sign of the expression inside the absolute value.

Case 1: x - 14 ≥ 0 In this case, the absolute value can be removed, so we have: x - 14 = 8 + 2x

Simplifying: 14 = x + 8 + 2x

Combining like terms: 14 = 3x + 8

Subtracting 8 from both sides: 6 = 3x

Dividing both sides by 3: x = 2

Case 2: x - 14 < 0 In this case, the expression inside the absolute value becomes its negation, so we have: -(x - 14) = 8 + 2x

Simplifying:

• x + 14 = 8 + 2x

Combining like terms: 14 = 3x + 8

Subtracting 8 from both sides: 6 = 3x

Dividing both sides by 3: x = 2

Therefore, the solution to the equation |x - 14| = 8 + 2x is x = 2.

3. |x^2 - 4x| = 3x

Once again, we consider two cases based on the sign of the expression inside the absolute value.

Case 1: x^2 - 4x ≥ 0 In this case, the absolute value can be removed, so we have: x^2 - 4x = 3x

Rearranging terms: x^2 - 7x = 0

Factoring out x: x(x - 7) = 0

Setting each factor equal to zero: x = 0 or x - 7 = 0

Therefore, the solutions for this case are x = 0 and x = 7.

Case 2: x^2 - 4x < 0 In this case, the expression inside the absolute value becomes its negation, so we have: -(x^2 - 4x) = 3x

Simplifying: -x^2 + 4x = 3x

Rearranging terms: -x^2 + 4x - 3x = 0

Combining like terms: -x^2 + x = 0

Factoring out x: x(-x + 1) = 0

Setting each factor equal to zero: x = 0 or -x + 1 = 0

Solving the second equation: -x + 1 = 0 x =