If x ∈ Z, solve 2 + 4x < 2x – 5 ≤ 3x. Also represent its solution on the number line.
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Given,
2+4x<2x−5≤3x⇒2+4x<2x−5 and 2x−5≤3x⇒4x−2x<−5−2 and 2x−3x≤5⇒2x<−7 and −x≤5⇒x<−72 and x≥−5⇒−5≤x<−722 + 4x \lt 2x - 5 \le 3x\\[0.5em] \Rightarrow 2 + 4x \lt 2x - 5 \text{ and } 2x - 5 \le 3x \\[0.5em] \Rightarrow 4x - 2x \lt - 5 - 2 \text{ and } 2x - 3x \le 5\\[0.5em] \Rightarrow 2x \lt -7 \text{ and } -x \le 5\\[0.5em] \Rightarrow x \lt -\dfrac{7}{2} \text{ and } x \ge -5\\[0.5em] \Rightarrow -5\le x \lt -\dfrac{7}{2}2+4x<2x−5≤3x⇒2+4x<2x−5 and 2x−5≤3x⇒4x−2x<−5−2 and 2x−3x≤5⇒2x<−7 and −x≤5⇒x<−27 and x≥−5⇒−5≤x<−27
Since x ∈ Z,
∴ Solution set = {-5, -4}.
The graph of the solution set is shown by thick dots on the number line.
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