Question

Не могу понять почему в уравнениях с модулями когда ответ минусовой иногда есть корень,а иногда нету. Пожалуйста,обьясните что к чему,сижу пару дней и не понимаю. "Арти"

Answer 1

Если |х| = 5
то будет корень
А если |x| = -5
то не будет

Answer 2

Understanding Equations with Absolute Values

Equations involving absolute values can sometimes have both positive and negative solutions, while other times they may only have one solution or no solution at all. This can be confusing, but there are certain principles that can help us understand why this happens.

Absolute Value Function

The absolute value function, denoted as |x|, returns the distance of a number x from zero on the number line. It always returns a non-negative value. For example, |3| = 3 and |-3| = 3.

Equations with Absolute Values

When solving equations with absolute values, we need to consider two cases: when the expression inside the absolute value is positive or zero, and when it is negative.

1. Expression inside the absolute value is positive or zero: - In this case, the absolute value equation simplifies to a regular equation without the absolute value. For example, |x - 2| = 4 becomes x - 2 = 4 or x - 2 = -4. - Solving these equations will give us the solutions. For the example above, the solutions are x = 6 and x = -2.

2. Expression inside the absolute value is negative: - When the expression inside the absolute value is negative, the absolute value equation has no solution. This is because the absolute value of a negative number is always positive, so there is no number that satisfies the equation. - For example, |x + 3| = -2 has no solution.

Example

Let's consider the equation |2x - 5| = 7.

1. Expression inside the absolute value is positive or zero: - When 2x - 5 is positive or zero, the equation becomes 2x - 5 = 7 or 2x - 5 = -7. - Solving these equations gives us x = 6 and x = -1.

2. Expression inside the absolute value is negative: - When 2x - 5 is negative, the equation has no solution.

Therefore, the solutions to the equation |2x - 5| = 7 are x = 6 and x = -1.

Conclusion

In summary, equations with absolute values can have different numbers of solutions depending on whether the expression inside the absolute value is positive or negative. When the expression is positive or zero, the equation simplifies to a regular equation. When the expression is negative, the equation has no solution. It's important to consider both cases when solving equations with absolute values.

I hope this explanation helps clarify your understanding of equations with absolute values. Let me know if you have any further questions!