Question

Solve each of the following system of equations in R.

\[5x - 7 < 3\left( x + 3 \right), 1 - \frac{3x}{2} \geq x - 4\]

 

Answer 1

\[5x - 7 < 3\left( x + 3 \right)\]
\[ \Rightarrow 5x - 7 < 3x + 9\]
\[ \Rightarrow 5x - 3x < 9 + 7\]
\[ \Rightarrow 2x < 16\]
\[ \Rightarrow x < 8\]
\[ \Rightarrow x \in \left( - \infty , 8 \right) . . . (i)\]
\[\text{ Also } , 1 - \frac{3x}{2} \geq x - 4\]
\[ \Rightarrow x - 4 \leq 1 - \frac{3x}{2}\]
\[ \Rightarrow x + \frac{3x}{2} \leq 1 + 4\]
\[ \Rightarrow \frac{2x + 3x}{2} \leq 5\]
\[ \Rightarrow 5x \leq 10 \]
\[ \Rightarrow x \leq 2 \]
\[ \Rightarrow x \in ( - \infty , 2] . . . (ii)\]
\[\text{ Hence, the solution of the given set of inequalities is theintersection of (i) and (ii) }  . \]
\[\left( - \infty , 8 \right) \cap ( - \infty , 2] = ( - \infty , 2]\]
\[\text{ Hence, the solution of the given set of inequalities is }  ( - \infty , 2] . \]