Question

Решить неравенство: (x-3)√x^2+4<=x^2-9

Answer 1

(x-3)√x^2+4<=x^2-9

x²-3x+4 ≤ x²-9 (√x²=x)

x²-x²-3x+4+9 ≤ 0

-3x+13  ≤ 0

-3x ≤ -13 | :(-3)

x ≥ 4,3 (3)

Answer 2

To solve the inequality, we'll follow these steps:

1. Isolate the radical expression on one side.
2. Square both sides to eliminate the square root (Note: This step may introduce extraneous solutions, so we need to check our solutions later).
3. Simplify the inequality.
4. Solve for the variable.
5. Verify the solutions to check for extraneous roots.

Let's go through the process step by step:

1. Isolate the radical expression on one side: Subtract x^2 and add 9 to both sides of the inequality to isolate the radical term:

   (x - 3)√(x^2 + 4) ≤ x^2 - 9 (x - 3)√(x^2 + 4) + 9 ≤ x^2

2. Square both sides to eliminate the square root: Squaring both sides of the inequality:

   [(x - 3)√(x^2 + 4) + 9]^2 ≤ x^4

   Now, we have to be careful, as squaring an inequality can introduce extraneous solutions.

3. Simplify the inequality: Expand the left side:

   (x - 3)^2(x^2 + 4) + 18(x - 3)√(x^2 + 4) + 81 ≤ x^4

   Simplify further:

   (x^2 - 6x + 9)(x^2 + 4) + 18(x - 3)√(x^2 + 4) + 81 ≤ x^4

   Now, let's combine the terms and move all terms to the left side of the inequality:

   x^4 - 6x^3 + 13x^2 - 6x - 36 + 18(x - 3)√(x^2 + 4) ≤ 0

4. Solve for the variable: At this point, solving the inequality explicitly may not be practical due to the presence of a radical term. Instead, we can make use of the sign of the expression within the square root to determine the valid intervals for x.

   The expression (x^2 + 4) inside the square root must be non-negative (≥ 0) for real solutions. So:

   x^2 + 4 ≥ 0

   The quadratic expression x^2 + 4 has no real roots because the discriminant (b^2 - 4ac) is negative for a = 1, b = 0, and c = 4. Therefore, it is always positive (greater than zero).

   As a result, the value of (x - 3)√(x^2 + 4) will always be non-negative.

   Hence, the inequality is satisfied for all real values of x.

5. Verify the solutions: Let's confirm our result by checking some values of x:

   Let's pick x = 0: (0 - 3)√(0^2 + 4) + 9 = (-3)√4 + 9 = (-3) * 2 + 9 = 3

   And x = 10: (10 - 3)√(10^2 + 4) + 9 = (7)√(100 + 4) + 9 = (7)√104 + 9 ≈ 32.5

   Both results satisfy the original inequality: 3 ≤ 0 is true, and 32.5 ≤ 91 is also true.

Thus, the inequality (x - 3)√(x^2 + 4) ≤ x^2 - 9 holds true for all real values of x.