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Mathematics

Solve the inequation, write down the solution set and represent it on a real number line:

3x−16<2x5−3≤−35+2x3x - 16 \lt \dfrac{2x}{5} - 3 \le -\dfrac{3}{5} + 2x; x ∈ R

Linear Inequations

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Answer

Given,

3x−16<2x5−3≤−35+2x3x - 16 \lt \dfrac{2x}{5} - 3 \le -\dfrac{3}{5} + 2x

Solving L.H.S. of the above equation :

⇒3x−16<2x5−3⇒3x−2x5<16−3⇒15x−2x5<13⇒15x−2x<13×5⇒13x<65⇒x<6513⇒x<5 ……..(1)\Rightarrow 3x - 16 \lt \dfrac{2x}{5} - 3 \\[1em] \Rightarrow 3x - \dfrac{2x}{5} \lt 16 - 3 \\[1em] \Rightarrow \dfrac{15x - 2x}{5} \lt 13 \\[1em] \Rightarrow 15x - 2x \lt 13 \times 5 \\[1em] \Rightarrow 13x \lt 65 \\[1em] \Rightarrow x \lt \dfrac{65}{13} \\[1em] \Rightarrow x \lt 5\text{ ……..(1)}

Solving R.H.S. of the above equation :

⇒2x5−3≤−35+2x⇒2x5−2x≤−35+3⇒2x−10x5≤−3+155⇒−8x5≤125⇒−8x≤12⇒8x≥−12⇒x≥−128⇒x≥−32 ……..(2)\Rightarrow \dfrac{2x}{5} - 3 \le -\dfrac{3}{5} + 2x \\[1em] \Rightarrow \dfrac{2x}{5} - 2x \le -\dfrac{3}{5} + 3\\[1em] \Rightarrow \dfrac{2x - 10x}{5} \le \dfrac{-3 + 15}{5}\\[1em] \Rightarrow \dfrac{-8x}{5} \le \dfrac{12}{5} \\[1em] \Rightarrow -8x \le 12 \\[1em] \Rightarrow 8x \ge -12 \\[1em] \Rightarrow x \ge -\dfrac{12}{8} \\[1em] \Rightarrow x \ge -\dfrac{3}{2} \text{ ……..(2)}

From equation (1) and (2), we get :

−32≤x<5-\dfrac{3}{2} \le x \lt 5

Solve the inequation, write down the solution set and represent it on a real number line: ICSE 2025 Improvement Maths Solved Question Paper.

Hence, solution set = {x : −32≤x<5-\dfrac{3}{2} \le x \lt 5, x ∈ R}.

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