Question

||8х+1|-2|=9 |4+|5х+2||=9 ||10х+7|-5|=13

Answer 1

1. $||8x+1|-2|=9\\|8x+1|-2=9\\|8x+1|-2=-9\\x_1=\frac{-3}{2}; x_2=\frac{5}{4}$  
2. $|4+|5x+2||=9\\|5x+2|=5\\5x+2=5; 5x+2=-5\\x_1=\frac{-7}{5} ;x_2=\frac{3}{5}$  
3. $||10x+7|-5|=13\\|10x+7|-5=13; |10x+7|-5=-13\\x_1=-\frac{5}{2}; x_2=\frac{11}{10}$

Answer 2

To solve the absolute value equations, we need to consider both the positive and negative cases of the expression within the absolute value bars and find the values of x that satisfy each case.

1. ||8x + 1| - 2| = 9

Case 1: (8x + 1) - 2 = 9 Solve for x: 8x + 1 - 2 = 9 8x - 1 = 9 8x = 10 x = 10/8 x = 5/4

Case 2: - (8x + 1) - 2 = 9 Solve for x:

• (8x + 1) - 2 = 9
• 8x - 1 - 2 = 9
• 8x - 3 = 9
• 8x = 12 x = 12/(-8) x = -3/2

So, the solutions for the first equation are x = 5/4 and x = -3/2.

2. |4 + |5x + 2|| = 9

Case 1: (5x + 2) = 5 Solve for x: 5x + 2 = 5 5x = 5 - 2 5x = 3 x = 3/5

Case 2: - (5x + 2) = 5 Solve for x:

• (5x + 2) = 5
• 5x - 2 = 5
• 5x = 5 + 2
• 5x = 7 x = 7/(-5) x = -7/5

So, the solutions for the second equation are x = 3/5 and x = -7/5.

3. ||10x + 7| - 5| = 13

Case 1: (10x + 7) - 5 = 13 Solve for x: 10x + 7 - 5 = 13 10x + 2 = 13 10x = 13 - 2 10x = 11 x = 11/10

Case 2: - (10x + 7) - 5 = 13 Solve for x:

• (10x + 7) - 5 = 13
• 10x - 7 - 5 = 13
• 10x - 12 = 13
• 10x = 13 + 12
• 10x = 25 x = 25/(-10) x = -5/2

So, the solutions for the third equation are x = 11/10 and x = -5/2.