Question

832. 1) ||x|- 3| = 0; 2)|5— |x|| = 0;3)||x| +3| = 0;4)|7 – |x|| = 0.​

Answer 1

$1) \ ||x| - 3| = 0\\|x| - 3 = 0\\|x| = 3\\x = \pm 3$

$2) \ |5 - |x|| = 0\\5 - |x| = 0\\-|x| = -5\\|x| = 5\\x = \pm 5$

$3) \ ||x| + 3| = 0\\|x| + 3 = 0\\|x| = -3$

Нет корней

$4) \ |7 - |x|| = 0\\7 - |x| = 0\\-|x| = -7\\|x| = 7\\x = \pm 7$

Answer 2

Let's solve each of the given equations one by one:

1. ||x| - 3| = 0:

To solve this equation, we can split it into two cases based on the inner absolute value.

Case 1: |x - 3| = 0: In this case, the expression inside the absolute value becomes zero. So we have: x - 3 = 0 x = 3

Case 2: |-(x - 3)| = 0: In this case, the expression inside the absolute value becomes zero when we negate it. So we have: -(x - 3) = 0 -x + 3 = 0 x = 3

Therefore, the solution to the equation ||x| - 3| = 0 is x = 3.

2. |5 - |x|| = 0:

Similar to the previous equation, we'll split this equation into two cases based on the inner absolute value.

Case 1: |x - 5| = 0: In this case, the expression inside the absolute value becomes zero. So we have: x - 5 = 0 x = 5

Case 2: |-(x - 5)| = 0: In this case, the expression inside the absolute value becomes zero when we negate it. So we have: -(x - 5) = 0 -x + 5 = 0 x = 5

Therefore, the solution to the equation |5 - |x|| = 0 is x = 5.

3. ||x| + 3| = 0:

Once again, let's split the equation into two cases based on the inner absolute value.

Case 1: |x + 3| = 0: In this case, the expression inside the absolute value becomes zero. So we have: x + 3 = 0 x = -3

Case 2: |-(x + 3)| = 0: In this case, the expression inside the absolute value becomes zero when we negate it. So we have: -(x + 3) = 0 -x - 3 = 0 x = -3

Therefore, the solution to the equation ||x| + 3| = 0 is x = -3.

4. |7 - |x|| = 0:

As before, we'll split this equation into two cases based on the inner absolute value.

Case 1: |x - 7| = 0: In this case, the expression inside the absolute value becomes zero. So we have: x - 7 = 0 x = 7

Case 2: |-(x - 7)| = 0: In this case, the expression inside the absolute value becomes zero when we negate it. So we have: -(x - 7) = 0 -x + 7 = 0 x = 7

Therefore, the solution to the equation |7 - |x|| = 0 is x = 7.

In summary:

1. ||x| - 3| = 0 has a solution x = 3.
2. |5 - |x|| = 0 has a solution x = 5.
3. ||x| + 3| = 0 has a solution x = -3.
4. |7 - |x|| = 0 has a solution x = 7.