Question

|2x+5|-|3-x|=0.5 2) |1-2x|+|3x+2|+|x|=5

Answer 1

Ответ:

3 *5^(2x-1) -2*5^(x-1)  = 0.2

3*5^2x* 5^(-1)   -  2 *5^x  * 5^(-1)  - 0.2=0

3/5  * 5^2x    -   2/5   *  5^x   -0.2=0

5^x= t 

3/5  t ²  -  2/5 t - 0.2 =0

0.6t²  - 0.4t -0.2=0          |:0,2

3t²  - 2t - 1 =0

D= 4 - 4*3*(-1) = 4+12 = 16 = 4²

t₁= (2-4) / (2*3) = -2/6 = -1/3

t₂ = (2+4)/6 = 6/6 =1

5^x= - 1/3  

5^x  = - 3^(-1) - нет вещественных корней

5^x = 1

5^x = 5^0

x=0

2)

3*5^(2x-1)-2*5^(x-1)=0,2

(3/5)*5^(2x)-(2/5)*5^x-0,2=0

5^x=t>0  ⇒

0,6t²-0,4t-0,2=0  I×5

3t²-2t-1-0  D=16

t₁=1  ⇒  5^x=1  x=0

t₂=-1/3  t₂∉

Объяснение:

Answer 2

To solve the equations |2x+5| - |3-x| = 0.5 and |1-2x| + |3x+2| + |x| = 5, we will break them down into cases based on the absolute values. Absolute values can lead to different cases because they can be either positive or negative. Let's start with the first equation:

Equation 1: |2x+5| - |3-x| = 0.5

Case 1: 2x + 5 is positive, and 3 - x is positive 2x + 5 - (3 - x) = 0.5 2x + 5 - 3 + x = 0.5 3x + 2 = 0.5

Case 2: 2x + 5 is positive, and 3 - x is negative 2x + 5 - (x - 3) = 0.5 2x + 5 - x + 3 = 0.5 x + 8 = 0.5

Case 3: 2x + 5 is negative, and 3 - x is positive -(2x + 5) - (3 - x) = 0.5 -2x - 5 - 3 + x = 0.5 -x - 8 = 0.5

Case 4: 2x + 5 is negative, and 3 - x is negative -(2x + 5) - (x - 3) = 0.5 -2x - 5 - x + 3 = 0.5 -3x - 2 = 0.5

Now, solve each of the equations from the four cases:

Case 1: 3x + 2 = 0.5 3x = -1.5 x = -0.5

Case 2: x + 8 = 0.5 x = -7.5

Case 3: -x - 8 = 0.5 -x = 8.5 x = -8.5

Case 4: -3x - 2 = 0.5 -3x = 2.5 x = -2.5/3

So, there are four possible solutions for Equation 1: x = -0.5, x = -7.5, x = -8.5, and x = -2.5/3.

Now, let's solve the second equation:

Equation 2: |1-2x| + |3x+2| + |x| = 5

Since there are three absolute values, we have eight possible cases:

1. 1 - 2x, 3x + 2, and x are all positive.
2. 1 - 2x and 3x + 2 are positive, while x is negative.
3. 1 - 2x and x are positive, while 3x + 2 is negative.
4. 3x + 2 and x are positive, while 1 - 2x is negative.
5. 1 - 2x is positive, while 3x + 2 and x are negative.
6. 3x + 2 is positive, while 1 - 2x and x are negative.
7. x is positive, while 1 - 2x and 3x + 2 are negative.
8. 1 - 2x, 3x + 2, and x are all negative.

For each of these cases, solve the equation. You should get multiple values of x. Some of them may be extraneous, so be sure to check your solutions in the original equation to verify their validity.