Question:
Solve each of the following system of equations in R.
7. 2x + 5 ≤ 0, x − 3 ≤ 0
Solution:
We have, $2 x+5 \leq 0$
$\Rightarrow 2 x \leq-5$
$\Rightarrow x \leq \frac{-5}{2}$
$\Rightarrow x \in\left(-\infty, \frac{-5}{2}\right] \quad \ldots(\mathrm{i})$
Also, $x-3 \leq 0$
$\Rightarrow x \leq 3$
$\Rightarrow x \in(-\infty, 3] \quad \ldots \quad$ (ii)
Thus, the solution of the given set of inequalities is the intersection of (i) and (ii).
$\left(-\infty, \frac{-5}{2}\right] \cap(-\infty, 3]=\left(-\infty, \frac{-5}{2}\right]$
Thus, the solution of the given set of inequalities is $\left(-\infty, \frac{-5}{2}\right]$.
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