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Mathematics

Solve the inequation :

−212+2x≤4x5≤43+2x-2\dfrac{1}{2} + 2x \le \dfrac{4x}{5} \le \dfrac{4}{3} + 2x, x ∈ W.

Graph the solution set on the number line.

Linear Inequations

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Answer

Given,

−212+2x≤4x5≤43+2x-2\dfrac{1}{2} + 2x \le \dfrac{4x}{5} \le \dfrac{4}{3} + 2x

Solving L.H.S. of the equation,

⇒−212+2x≤4x5⇒−52+2x≤4x5⇒2x−4x5≤52⇒10x−4x5≤52⇒6x5≤52⇒x≤52×56⇒x≤2512⇒x≤2112........(i)\Rightarrow -2\dfrac{1}{2} + 2x \le \dfrac{4x}{5} \\[1em] \Rightarrow -\dfrac{5}{2} + 2x \le \dfrac{4x}{5} \\[1em] \Rightarrow 2x - \dfrac{4x}{5} \le \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{10x - 4x}{5} \le \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{6x}{5} \le \dfrac{5}{2} \\[1em] \Rightarrow x \le \dfrac{5}{2} \times \dfrac{5}{6} \\[1em] \Rightarrow x \le \dfrac{25}{12} \\[1em] \Rightarrow x \le 2\dfrac{1}{12} ……..(i)

Solving R.H.S. of the equation,

⇒4x5≤43+2x⇒4x5−2x≤43⇒4x−10x5≤43⇒−6x5≤43⇒6x5≥−43⇒x≥−43×56⇒x≥−109⇒x≥−119.......(ii)\Rightarrow \dfrac{4x}{5} \le \dfrac{4}{3} + 2x \\[1em] \Rightarrow \dfrac{4x}{5} - 2x \le \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{4x - 10x}{5} \le \dfrac{4}{3} \\[1em] \Rightarrow -\dfrac{6x}{5} \le \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{6x}{5} \ge -\dfrac{4}{3} \\[1em] \Rightarrow x \ge -\dfrac{4}{3} \times \dfrac{5}{6} \\[1em] \Rightarrow x \ge -\dfrac{10}{9} \\[1em] \Rightarrow x \ge -1\dfrac{1}{9} …….(ii)

From (i) and (ii) we get,

−119≤x≤2112-1\dfrac{1}{9} \le x \le 2\dfrac{1}{12}

Since, x ∈ W

∴ Solution set = {0, 1, 2}.

Solution on the number line is :

Solve the inequation -2(1/2) + 2x ≤ (4x/5) ≤ 4/3 + 2x, x ∈ W. Graph the solution set on the number line. Linear Inequations, Concise Mathematics Solutions ICSE Class 10.

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