Question

solve each inequality. Graph and check your solution. (if no solution write no solution if all real numbers write all real numbers.)5n + 3 > equal to 7(n-3)6 - 2p > equal to -33 (y - 4) < 5 (y + 2)

Answer 1

The first given inequality is

`5n+3\ge7(n-3)`

First, we use the distributive property

`5n+3\ge7n-21`

Second, we subtract 7n on each side

`\begin{gathered} 5n+3-7n\ge7n-21-7n \\ -2n+3\ge-21 \end{gathered}`

Third, we subtract 3 on each side

`\begin{gathered} -2n+3-3\ge-21-3 \\ -2n\ge-24 \end{gathered}`

At last, we divide the inequality by -2, which changes the inequality sign

`\begin{gathered} (-2n)/(-2)\leq(-24)/(-2) \\ n\leq12 \end{gathered}`

Therefore, the solution is all real numbers less than or equal to 12.

We repeat the process for the other inequalities.

`\begin{gathered} 6-2p\ge-3 \\ 6-2p-6\ge-3-6 \\ -2p\ge-9 \\ (-2p)/(-2)\leq(-9)/(-2) \\ p\leq(9)/(2) \end{gathered}`

The solution is all real numbers less than or equal to 9/2.

The last inequality would be

`\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 \\ (-2y)/(-2)>(22)/(-2) \\ y>-11 \end{gathered}`

The solution is all real numbers greater than -11.