First degree inequations

Solve and find for which values of $x$ for the following inequations:

  1. $2x-3 < 1$

  2. $x > 3(x+1)$

  3. $-\dfrac{2}{3}(3x-6) \geqslant 2x-1$

We are going to solve 3 inequations step by step giving ,at the end, the values of $x$ for which the inequation is satisfied.

  1. $2x-3 < 1 \Rightarrow 2x < 1+3 \Rightarrow 2x < 4 \Rightarrow x < \dfrac{4}{2} \Rightarrow x < 2$.

  2. $x > 3(x+1) \Rightarrow x > 3x+3 \Rightarrow x- 3x > 3 \Rightarrow -2x > 3 \Rightarrow x < \dfrac{3}{-2} $.

  3. $-\dfrac{2}{3}(3x-6) \geqslant 2x-1 \Rightarrow -\dfrac{2}{3}\cdot 3x + \dfrac{2}{3} \cdot 6 \geqslant 2x-1 \Rightarrow$ $\Rightarrow-2x + 4 \geqslant 2x-1 \Rightarrow 4+1 \geqslant 2x + 2x \Rightarrow 5 \geqslant 4x \Rightarrow \dfrac{5}{4} \geqslant x$.

  1. $x < 2$

  2. $ x > \dfrac{3}{-2} $

  3. $ x \leqslant \dfrac{5}{4}$

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