Question

Решите неравенства: 1) | x -    | < 3 2) | x -     | * |   | < 7

Answer 1

Всё подробно написала в решении.

Answer 2

1. |x - | | < 3:

To solve this inequality, we'll break it down into cases:

Case 1: x - |x| is non-negative or zero (x - |x| ≥ 0) In this case, the absolute value of x is the same as x because x is non-negative or zero. So, we can write the inequality as: x - x < 3

This simplifies to 0 < 3, which is true. So, all values of x that satisfy this case are part of the solution.

Case 2: x - |x| is negative (x - |x| < 0) In this case, the absolute value of x is the opposite of x because x is negative. So, we can write the inequality as: x + x < 3

This simplifies to 2x < 3, and if we divide by 2 (keeping in mind to reverse the inequality because we're dividing by a negative number), we get: x > 3/2

So, all values of x that satisfy this case are x > 3/2.

Now, let's combine the solutions from both cases:

For case 1, we have all values of x.

For case 2, we have x > 3/2.

So, the solution to the inequality is: x ∈ (-∞, 3/2) ∪ (3/2, ∞)

2. |x - | | * | | < 7:

To solve this inequality, we'll also break it down into cases:

Case 1: x - |x| is non-negative or zero (x - |x| ≥ 0) In this case, the absolute value of x is the same as x, so the inequality becomes: x * x < 7

This simplifies to x^2 < 7. To find the solution, take the square root, but remember to consider both the positive and negative square roots since we don't know the sign of x yet: -√7 < x < √7

Case 2: x - |x| is negative (x - |x| < 0) In this case, the absolute value of x is the opposite of x, so the inequality becomes: x * (-x) < 7

This simplifies to -x^2 < 7. To solve for x, multiply both sides by -1 and reverse the inequality: x^2 > -7

Since x^2 is always non-negative, all real values of x satisfy this inequality.

Now, let's combine the solutions from both cases:

For case 1, we have -√7 < x < √7.

For case 2, all real values of x satisfy the inequality.

So, the solution to the inequality is: x ∈ (-∞, ∞)