The first given inequality is
[tex]5n+3\ge7(n-3)[/tex]
First, we use the distributive property
[tex]5n+3\ge7n-21[/tex]
Second, we subtract 7n on each side
[tex]\begin{gathered} 5n+3-7n\ge7n-21-7n \\ -2n+3\ge-21 \end{gathered}[/tex]
Third, we subtract 3 on each side
[tex]\begin{gathered} -2n+3-3\ge-21-3 \\ -2n\ge-24 \end{gathered}[/tex]
At last, we divide the inequality by -2, which changes the inequality sign
[tex]\begin{gathered} \frac{-2n}{-2}\leq\frac{-24}{-2} \\ n\leq12 \end{gathered}[/tex]
Therefore, the solution is all real numbers less than or equal to 12.
We repeat the process for the other inequalities.
[tex]\begin{gathered} 6-2p\ge-3 \\ 6-2p-6\ge-3-6 \\ -2p\ge-9 \\ \frac{-2p}{-2}\leq\frac{-9}{-2} \\ p\leq\frac{9}{2} \end{gathered}[/tex]
The solution is all real numbers less than or equal to 9/2.
The last inequality would be
[tex]\begin{gathered} 3(y-4)<5(y+2) \\ 3y-12<5y+10 \\ 3y-12-5y<5y+10-5y \\ -2y-12<10 \\ -2y-12+12<10+12 \\ -2y<22 \\ \frac{-2y}{-2}>\frac{22}{-2} \\ y>-11 \end{gathered}[/tex]
The solution is all real numbers greater than -11.
