Question

Решите неравенство: √x+2 > √4-x

Answer 1

Решение на фотографии

Answer 2

To solve this inequality, we can start by squaring both sides of the inequality. However, we must be careful to check for any extraneous solutions at the end, since squaring both sides can sometimes introduce solutions that do not satisfy the original inequality.

√x+2 > √4-x

Square both sides:

(√x+2)² > (√4-x)²

Simplifying:

x + 4 + 2√x+2 > 4 - x

Isolate the radical term:

2√x+2 > -x

Square both sides again:

4x+8 > x²

Rearrange:

x² - 4x - 8 < 0

Using the quadratic formula:

x = [4 ± √(16+32)]/2 = 2 ± 2√3

Therefore, the solutions to the inequality are:

2 - 2√3 < x < 2 + 2√3

However, we need to check for extraneous solutions that were introduced by squaring both sides. We can see that x = 2 - 2√3 is not a valid solution, because plugging it back into the original inequality results in a false statement:

√(2-2√3+2) > √(4-(2-2√3)) √4-2√3 > √2+2√3

which is not true, since 2√3 > √2.

Therefore, the solution to the inequality is:

2 - 2√3 < x < 2 + 2√3, x ≠ 2 - 2√3