Question

Решить |4x-1|=(3x-1)^-1

Answer 1

При 4x-1≥0, т.е х≥¼
и x≠⅓
получаем:
(4х-1)=1/(3х-1)

(4х-1)(3х-1)=1
12х²-7х+1-1=0
х(12х-7)=0
х=0 ( не подходит, т к. х≥¼ )
х=7/12

при 4x-1<0, х<¼
получаем:
(4х-1)= -1/(3х-1)

(4х-1)(3х-1)= -1

12х²-7х+1= -1

12х²-7х+2=0
D = b² - 4ac = (-7)2 - 4·12·2 = 49 - 96 = -47<0
решений нет

Поэтому
ответ х=7/12

Answer 2

To solve the equation |4x - 1| = (3x - 1)^(-1), we need to consider two cases: when (3x - 1)^(-1) is positive and when it's negative or zero. Let's break down the steps for both cases:

Case 1: (3x - 1)^(-1) is positive In this case, the equation |4x - 1| = (3x - 1)^(-1) remains as it is.

Case 2: (3x - 1)^(-1) is negative or zero For (3x - 1)^(-1) to be negative or zero, we must have 3x - 1 ≠ 0. In other words, x ≠ 1/3.

Now, let's proceed with solving each case:

Case 1: (3x - 1)^(-1) is positive

Since the absolute value |4x - 1| is always non-negative (greater than or equal to zero), we can safely ignore the absolute value sign and proceed with the equation as is:

(3x - 1)^(-1) = 4x - 1

To eliminate the fraction, we can take the reciprocal of both sides:

1/(3x - 1) = 4x - 1

Now, we can solve for x:

1 = (4x - 1)(3x - 1)

Expand the right side:

1 = 12x^2 - 7x + 1

Now, move all terms to one side:

12x^2 - 7x = 0

Factor out x:

x(12x - 7) = 0

Now, set each factor to zero and solve for x:

1. x = 0

2. 12x - 7 = 0 12x = 7 x = 7/12

So, in Case 1, we have two potential solutions: x = 0 and x = 7/12.

Case 2: (3x - 1)^(-1) is negative or zero

In this case, we need to consider x ≠ 1/3. However, the equation |4x - 1| = (3x - 1)^(-1) is not applicable in this scenario because (3x - 1)^(-1) is either zero or negative, which means the left side would be non-negative (positive or zero), while the right side would be negative or zero. Since they can't be equal, there are no valid solutions for x in this case.

Final Solution:

The solutions to the equation |4x - 1| = (3x - 1)^(-1) are x = 0 and x = 7/12.