Question

√(x-5) + √(10-x) < 3

Answer 1

-/(x-5)^2<(3--/(10-x))^2
x-5 < 9-6-/10-x)+10-x
x-5 < 19-6-/10-x)-x
(6-/10-x)^2 < (24 - 2x ) ^2
36 (10-x)< 576-96x +4x
360- 36 x < 576 -92x
56x < 216
x < 216/56
x < 27/7
решил как смог !!!

Answer 2

To solve the inequality √(x-5) + √(10-x) < 3, follow these steps:

1. Start by isolating one of the square root terms on one side of the inequality. Let's isolate √(x-5):

   √(x-5) < 3 - √(10-x)

2. Now, square both sides of the inequality to eliminate the square root:

   (x-5) < (3 - √(10-x))^2

3. Expand the right side:

   x - 5 < 9 - 6√(10-x) + (10-x)

4. Simplify the terms:

   x - 5 < 19 - 6√(10-x) - x

5. Combine like terms:

   2x < 24 - 6√(10-x)

6. Divide both sides by 2:

   x < 12 - 3√(10-x)

7. Square both sides of the inequality again:

   x^2 < (12 - 3√(10-x))^2

8. Expand the right side:

   x^2 < 144 - 72√(10-x) + 9(10-x)

9. Simplify the terms:

   x^2 < 234 - 72√(10-x)

10. Subtract 234 from both sides:

x^2 - 234 < - 72√(10-x)

11. Divide both sides by -72 (remember to flip the inequality sign since you're dividing by a negative number):

(x^2 - 234)/-72 > √(10-x)

12. Square both sides of the inequality yet again (remember that squaring both sides of an inequality can introduce extraneous solutions, so you need to check the solutions later):

((x^2 - 234)/-72)^2 > 10 - x

13. Expand the left side:

(x^2 - 234)^2 / 72^2 > 10 - x

14. Simplify the left side:

(x^4 - 468x^2 + 54756) / 5184 > 10 - x

15. Multiply both sides by 5184:

x^4 - 468x^2 + 54756 > 51840 - 5184x

16. Move all terms to one side:

x^4 - 468x^2 + 5184x - 54756 > 0

Now, you have a quartic inequality that you can solve to find the range of values for x that satisfy the original inequality. However, solving quartic inequalities can be complex and may involve numerical methods or graphing software. The solution process can become quite intricate, so it's recommended to use a calculator or software to solve for the specific range of x values that satisfy the inequality.