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If x ∈ R, solve 2x−3≥x+1−x3>25x2x - 3 \ge x+\dfrac{1−x}{3} \gt \dfrac{2}{5}x. Also represent the solution on the number line.

Linear Inequations

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Answer

Given,

2x−3≥x+1−x3>25x2x-3 \ge x + \dfrac{1-x}{3} \gt \dfrac{2}{5}x

Solving left side,

2x−3≥x+1−x3⇒2x−3≥3x+1−x3⇒2x−3≥2x+13⇒3(2x−3)≥2x+1⇒6x−9≥2x+1⇒6x−2x≥1+9⇒4x≥10⇒x≥104⇒x≥522x-3 \ge x + \dfrac{1-x}{3} \\[0.5em] \Rightarrow 2x - 3 \ge \dfrac{3x +1 -x}{3} \\[0.5em] \Rightarrow 2x -3 \ge \dfrac{2x + 1}{3} \\[0.5em] \Rightarrow 3 (2x -3) \ge 2x+1 \\[0.5em] \Rightarrow 6x - 9 \ge 2x + 1 \\[0.5em] \Rightarrow 6x - 2x \ge 1+ 9 \\[0.5em] \Rightarrow 4x \ge 10 \\[0.5em] \Rightarrow x \ge \dfrac{10}{4} \\[0.5em] \Rightarrow x \ge \dfrac{5}{2}

Solving right side,

x+1−x3>25x⇒3x+1−x3>25x⇒2x+13>25x⇒5(2x+1)>6x⇒10x+5>6x⇒10x−6x>−5⇒4x>−5⇒x>−54x + \dfrac{1-x}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow \dfrac{3x + 1 - x}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow \dfrac{2x + 1}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow 5(2x +1) \gt 6x \\[0.5em] \Rightarrow 10x + 5 \gt 6x \\[0.5em] \Rightarrow 10x - 6x \gt -5 \\[0.5em] \Rightarrow 4x \gt -5 \\[0.5em] \Rightarrow x \gt -\dfrac{5}{4}

From left side we get x≥52x \ge \dfrac{5}{2} and from right side we get x>−54x \gt -\dfrac{5}{4}

∴ x≥52x \ge \dfrac{5}{2}

∴ Solution set = {x : x ∈ R, x ≥52\ge \dfrac{5}{2} }

The graph of the inequation is represented by thick black line starting from 52\dfrac{5}{2} (including 52\dfrac{5}{2}).

If x ∈ R, solve 2x - 3 ≥ x + (1−x)/(3) > (2x/5). Also represent the solution on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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