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Prove that if $\begin{eqnarray} a_1&\leq& b_1\leq c_1,\\ a_2&\leq& b_2\leq c_2,\\ &\vdots&\\ a_N&\leq& b_N\leq c_N, \end{eqnarray}$ then $\sum_{i=1}^Na_i\leq \sum_{i=1}^Nb_i\leq \sum_{i=1}^Nc_i.$
Prove that if $a_1\leq b_1\leq c_1$ and $a_2\leq b_2\leq c_2$ , then $a_1+a_2\leq b_1+b_2\leq c_1+c_2$ .
Prove that if $a_1\leq b_1$ and $a_2\leq b_2$ , then $a_1+a_2\leq b_1+b_2$ .
Prove that if $a=bc$ , $b\gt 0$ and $c\gt 1$ , then $a\gt b$ .
Prove that if $a\gt 0$ and $b\gt 0$ , then $a+b\gt a$ and $a+b\gt b$ .
Prove that if $0\lt a\lt b$ , then $a^2\lt \frac{1}{3}\left(a^2+ab+b^2\right)\lt b^2$ .
Prove that if $0\lt x_1\lt x_2$ , then $x^2_1\lt x^2_2$ .
Solve for $x$ in $5-3x\leq 8+5x$ .