x-2-4x-x-3-

Question Number 139695 by john_santu last updated on 30/Apr/21

∣ x^{2} - 4 x ∣ + ∣ x ∣ ⩾ 3

Answered by MJS_new last updated on 30/Apr/21

∣ x (x - 4) ∣ + ∣ x ∣= 3

squaring

x^{2} {(1 + ∣ x - 4 ∣)}^{2} = 9

2 x^{2} ∣ x - 4 ∣= - x^{4} + 8 x^{3} - 17 x^{2} + 9

squaring & transforming

x^{8} - 16 x^{7} + 94 x^{6} - 240 x^{5} + 207 x^{4} + 144 x^{3} - 306 x^{2} + 81 = 0

x = t + 2

t^{8} - 18 t^{6} - 8 t^{5} + 87 t^{4} + 72 t^{3} - 74 t^{2} - 168 t - 135 = 0

(t^{2} - t - 9) (t^{2} - t - 3) (t^{2} + t - 5) (t^{2} + t + 1) = 0

this is easy to solve but only 2 solutions fit

the original equation

x_{1} = \frac{5 - \sqrt{37}}{2} \land x_{2} = \frac{5 - \sqrt{13}}{2}

\Rightarrow

x ⩽ \frac{5 - \sqrt{37}}{2} \lor x ⩾ \frac{5 - \sqrt{13}}{2}

Answered by mr W last updated on 30/Apr/21

f o r x ⩽ 0 :

x^{2} - 4 x - x ⩾ 3

x^{2} - 5 x - 3 ⩾ 0

x ⩽ \frac{5 - \sqrt{37}}{2}

f o r 0 ⩽ x ⩽ 4 :

- x^{2} + 4 x + x ⩾ 3

x^{2} - 5 x + 3 ⩽ 0

\frac{5 - \sqrt{13}}{2} ⩽ x ⩽ \frac{5 + \sqrt{13}}{2}

\Rightarrow \frac{5 - \sqrt{13}}{2} ⩽ x ⩽ 4

f o r x ⩾ 4 :

x^{2} - 4 x + x ⩾ 3

x^{2} - 3 x - 3 ⩾ 0

x ⩾ \frac{3 + \sqrt{21}}{2}

\Rightarrow x ⩾ 4

s u m m a r y :

x ⩽ \frac{5 - \sqrt{37}}{2} o r x ⩾ \frac{5 - \sqrt{13}}{2}

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