What is (sqrt(|x+4|^2-25))*(sqrt(x^2-9))<= 0 ?

The solution to (sqrt(|x+4|^2-25))*(sqrt(x^2-9))<= 0 is x=-9\lor x=3

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(sqrt(|x+4|^2-25))*(sqrt(x^2-9))<= 0

(∣ x + 4 ∣^{2} - 25) \cdot (x^{2} - 9) \leq 0

x = - 9 or x = 3

Interval Notation

x = - 9 \cup x = 3

Solution steps

∣ x + 4 ∣^{2} - 25 x^{2} - 9 \leq 0

Apply radical rule:

a b = a \cdot b,

assuming

a \geq 0, b \geq 0

∣ x + 4 ∣^{2} - 25 x^{2} - 9 = (∣ x + 4 ∣^{2} - 25) (x^{2} - 9)

(∣ x + 4 ∣^{2} - 25) (x^{2} - 9) \leq 0

Square both sides

((∣ x + 4 ∣^{2} - 25) (x^{2} - 9))^{2} \leq 0^{2}

Simplify

x^{4} - 18 x^{2} + 8 x^{3} - 72 x + 81 \leq 0

x^{4} - 18 x^{2} + 8 x^{3} - 72 x + 81 \leq 0 : - 9 \leq x \leq - 3 or 1 \leq x \leq 3

- 9 \leq x \leq - 3 or 1 \leq x \leq 3

Find singularity points

Find non-negative values for radicals:

x \leq - 9 or x \geq 3

Combine the intervals

(- 9 \leq x \leq - 3 or 1 \leq x \leq 3) and (x \leq - 9 or x \geq 3)

Merge Overlapping Intervals

x = - 9, x = 3

x = - 9 or x = 3

5 (5 x + 7) = 3 (3 x + 4) + 55

8 = x 3

(2 x^{3} y^{- 3})^{- 2}

9 - 6 x + x^{2}

∣ x - 5 ∣ \geq 4

What is (sqrt(|x+4|^2-25))*(sqrt(x^2-9))<= 0 ?
The solution to (sqrt(|x+4|^2-25))*(sqrt(x^2-9))<= 0 is x=-9\lor x=3