x-2-3x-2-x-0-find-range-of-values-of-x-agreeing-with-above-inequality-

Question Number 16589 by ajfour last updated on 24/Jun/17

x^{2} - ∣ 3 x + 2 ∣ + x ⩾ 0

find range of values of x agreeing

with above inequality .

Answered by mrW1 last updated on 24/Jun/17

if 3 x + 2 ⩾ 0 or x ⩾ - \frac{2}{3} (\approx - 0.67)

x^{2} - 3 x - 2 + x ⩾ 0

x^{2} - 2 x + 1 - 3 ⩾ 0

{(x - 1)}^{2} ⩾ 3

x - 1 ⩾ \sqrt{3} \Rightarrow x ⩾ 1 + \sqrt{3}

or

x - 1 ⩽ - \sqrt{3} \Rightarrow x ⩽ 1 - \sqrt{3} (\approx - 0.73) ≱ - \frac{2}{3}

\Rightarrow\Rightarrow x ⩾ 1 + \sqrt{3}

if 3 x + 2 < 0 or x < - \frac{2}{3} (\approx - 0.67)

x^{2} + 3 x + 2 + x ⩾ 0

x^{2} + 4 x + 2 ⩾ 0

{(x + 2)}^{2} ⩾ 2

x + 2 ⩾ \sqrt{2} \Rightarrow x ⩾ - 2 + \sqrt{2} (\approx - 0.59) ≮ - \frac{2}{3}

or

x + 2 ⩽ - \sqrt{2} \Rightarrow x ⩽ - 2 - \sqrt{2} (\approx - 3.41)

\Rightarrow\Rightarrow x ⩽ - 2 - \sqrt{2}

\Rightarrow x \in (- \infty, - 2 - \sqrt{2}] \land [1 + \sqrt{3}, + \infty)

Commented by ajfour last updated on 24/Jun/17

absolutely right Sir . Thanks .

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