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Mathematics

Solve the following inequation, write the solution set and represent it on the real number line.

−1<2x−33−x5≤1,x∈R-1 \lt \dfrac{2x - 3}{3} - \dfrac{x}{5} \le 1, x \in R

Linear Inequations

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Answer

Solve the following inequation, write the solution set and represent it on the real number line. ICSE 2026 Maths Solved Question Paper.

Given,

−1<2x−33−x5≤1,x∈R-1 \lt \dfrac{2x - 3}{3} - \dfrac{x}{5} \le 1, x \in R

Solving L.H.S of the inequation,

⇒−1<2x−33−x5⇒−1<10x−15−3x15⇒−1<7x−1515⇒−15<7x−15⇒−15+15<7x⇒0<7x⇒x>0.\Rightarrow -1 \lt \dfrac{2x - 3}{3} - \dfrac{x}{5} \\[1em] \Rightarrow -1 \lt \dfrac{10x - 15 - 3x}{15} \\[1em] \Rightarrow -1 \lt \dfrac{7x - 15}{15} \\[1em] \Rightarrow -15 \lt 7x - 15 \\[1em] \Rightarrow -15 + 15 \lt 7x \\[1em] \Rightarrow 0 \lt 7x \\[1em] \Rightarrow x \gt 0.

Solving R.H.S of the inequation,

⇒2x−33−x5≤1⇒10x−15−3x15≤1⇒7x−1515≤1⇒7x−15≤15⇒7x≤15+15⇒7x≤30⇒x≤307.\Rightarrow \dfrac{2x - 3}{3} - \dfrac{x}{5} \le 1 \\[1em] \Rightarrow \dfrac{10x - 15 - 3x}{15} \le 1 \\[1em] \Rightarrow \dfrac{7x - 15}{15} \le 1 \\[1em] \Rightarrow 7x - 15 \le 15 \\[1em] \Rightarrow 7x \le 15 + 15 \\[1em] \Rightarrow 7x \le 30 \\[1em] \Rightarrow x \le \dfrac{30}{7}.

∴ 0 < x ≤ 307\dfrac{30}{7}

Solution set = (0,307]\Big(0, \dfrac{30}{7}\Big]

Hence, solution set = (0,307]\Big(0, \dfrac{30}{7}\Big].

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