Question

Имеет ли уравнение 5x-31=|x-3| корень, меньший 2? Если нет, то почему, если имеет, то какой?

Answer 1

5x-31=|x-3|
х-3=0
х=3
____________________3_______________
1)x<3
  5x-31=-(x-3)
  5x-31=-x+3
  5x+x=3+31
  6x=34 |:6
  x=17/3
  x=5 2/3∉(-∞;3)
  

2)x≥3
  5x-31=x-3
  5x-x= 31-3
  4x=29 |:4
  x=29/4
  x=7 1/4∈[3;+∞)
  x=7,25 - единственный корень уравнения
  Уравнение не имеет корней меньших 2, т.к. показано, что на (-∞;3) корней не существует.

Answer 2

Checking the Equation for a Root Less Than 2

To determine if the equation 5x-31=|x-3| has a root less than 2, we can analyze the equation and solve for the value of x.

Solving the Equation

First, let's solve the equation step by step.

The given equation is: 5x-31=|x-3|

To solve for x, we can consider two cases: 1. When x-3 is non-negative, i.e., x-3 ≥ 0 2. When x-3 is negative, i.e., x-3 < 0

Case 1: x-3 ≥ 0

In this case, the absolute value |x-3| is equal to (x-3). So, the equation becomes: 5x-31=x-3

Solving for x: 5x-x=31-3 4x=28 x=7

Case 2: x-3 < 0

In this case, the absolute value |x-3| is equal to -(x-3). So, the equation becomes: 5x-31=-(x-3)

Solving for x: 5x-31=-x+3 5x+x=31+3 6x=34 x=17/3

Conclusion

From the above analysis, we find that the equation 5x-31=|x-3| has two potential solutions: 1. x=7 2. x=17/3

To determine if either of these solutions is less than 2, we can compare them to 2.

Comparing Solutions to 2

Comparing the potential solutions to 2: - 7 is greater than 2. - 17/3 is also greater than 2.

Therefore, neither of the solutions for x is less than 2.

Final Answer

The equation 5x-31=|x-3| does not have a root less than 2.

This conclusion is based on the analysis of the equation and the comparison of the potential solutions to the given condition.

I hope this helps! If you have any further questions, feel free to ask.