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junior college 1 | H2 Maths
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help to solve this inequality !
Date Posted: 3 years ago
Views: 525
x⁴ ≤ 9
x⁴ - 9 ≤ 0
(x² - 3)(x² + 3) ≤ 0
Since x² + 3 > 0 for all real x ,
Then x² - 3 ≤ 0
(x - √3)(x + √3) ≤ 0
Drawing the number line and looking at the sign,
For -√3 < x < √3, (x - √3)(x + √3) < 0
When x = ±√3, (x - √3)(x + √3) = 0
When x > √3 or x < -√3, (x - √3)(x + √3) > 0
So,
-√3 ≤ x ≤ √3
x⁴ - 9 ≤ 0
(x² - 3)(x² + 3) ≤ 0
Since x² + 3 > 0 for all real x ,
Then x² - 3 ≤ 0
(x - √3)(x + √3) ≤ 0
Drawing the number line and looking at the sign,
For -√3 < x < √3, (x - √3)(x + √3) < 0
When x = ±√3, (x - √3)(x + √3) = 0
When x > √3 or x < -√3, (x - √3)(x + √3) > 0
So,
-√3 ≤ x ≤ √3
Alternatively,
x⁴ ≤ 9
(x²)² ≤ 3²
→ |x²|≤ 3
But since x² ≥ 0 for all real x, then |x²| = x²
So x² ≤ 3
x² ≤ (√3)²
|x| ≤ √3
-√3 ≤ x ≤ √3
x⁴ ≤ 9
(x²)² ≤ 3²
→ |x²|≤ 3
But since x² ≥ 0 for all real x, then |x²| = x²
So x² ≤ 3
x² ≤ (√3)²
|x| ≤ √3
-√3 ≤ x ≤ √3
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