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Solve the inequation 3 ≥ x−42+x3≥2\dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2, x ∈ I (integers). Graph the solution on a real number line.

Linear Inequations

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Answer

Given inequation : 3 ≥ x−42+x3≥2\dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2

Solving L.H.S. of the inequation :

⇒3≥x−42+x3⇒3≥3(x−4)+2x6⇒3≥3x−12+2x6⇒18≥5x−12⇒5x≤18+12⇒5x≤30⇒x≤305⇒x≤6 ……….(1)\Rightarrow 3 \ge \dfrac{x - 4}{2} + \dfrac{x}{3} \\[1em] \Rightarrow 3 \ge \dfrac{3(x - 4) + 2x}{6} \\[1em] \Rightarrow 3 \ge \dfrac{3x - 12 + 2x}{6} \\[1em] \Rightarrow 18 \ge 5x - 12 \\[1em] \Rightarrow 5x \le 18 + 12 \\[1em] \Rightarrow 5x \le 30 \\[1em] \Rightarrow x \le \dfrac{30}{5} \\[1em] \Rightarrow x \le 6 \text{ ……….(1)}

Solving R.H.S. of the inequation :

⇒x−42+x3≥2⇒3(x−4)+2x6≥2⇒3x−12+2x6≥2⇒3x−12+2x≥12⇒5x≥24⇒x≥245⇒x≥445 ………(2)\Rightarrow \dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2 \\[1em] \Rightarrow \dfrac{3(x - 4) + 2x}{6} \ge 2 \\[1em] \Rightarrow \dfrac{3x - 12 + 2x}{6} \ge 2 \\[1em] \Rightarrow 3x - 12 + 2x \ge 12 \\[1em] \Rightarrow 5x \ge 24 \\[1em] \Rightarrow x \ge \dfrac{24}{5} \\[1em] \Rightarrow x \ge 4\dfrac{4}{5} \text{ ………(2)}

From (1) and (2),

445≤x≤64\dfrac{4}{5} \le x \le 6 and x ∈ I.

Solve the inequation 3 ≥ x - 4/2 + x/3 ≥ 2, x ∈ I (integers). Graph the solution on a real number line. Model Paper 5, Concise Mathematics Solutions ICSE Class 10.

Solution set = {5, 6}.

Hence, solution set = {5, 6}.

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