|x^2-4|<= 3
1) Let's solve in ℝ: |x2 -4| ≤ 3
|A| ≤ B is equivalent to -B ≤ A ≤ B
2) Let's solve in ℝ: x2 -4 ≤ 3
ⓘ We add 4 to both sides
x2 ≤ 3 + 4
ⓘ Start from 3 and move forward 4 on the number line: 3 + 4 = 7
x2 ≤ 7
An even power only depends on the absolute value: x2 ≤ 7 is equivalent to |x| ≤ 7
|A| ≤ B is equivalent to -B ≤ A ≤ B
-7 ≤ x ≤ 7
Solution obtained: x ∈ [-7, 7]
3) Let's solve in ℝ: x2 -4 ≥ -3
ⓘ We add 4 to both sides
x2 ≥ -3 + 4
ⓘ Start from -3 and move forward 4 on the number line, going past zero: -3 + 4 = 1
x2 ≥ 1
An even power only depends on the absolute value: x2 ≥ 1 is equivalent to |x| ≥ 1
|A| ≥ B is equivalent to: A ≥ B or A ≤ -B
Let's build the table of the different terms to determine the global solution:
x-∞ -7 -1 1 7 +∞
x2 -4 ≥ -3✅✅✅✅❌✅✅✅✅
x2 -4 ≤ 3❌✅✅✅✅✅✅✅❌
Global solution❌✅✅✅❌✅✅✅❌
▶Solution found: x ∈ [-7, -1] ∪ [1, 7]
4) The numeric value of: x ∈ [-7, -1] ∪ [1, 7] equals:
▷ x ∈ [≈ -2.6458, -1] ∪ [1, ≈ 2.6458]