Solve each inequality separately.

[latex]\displaystyle \begin{array}{r}2y+7<13\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-3y-2\lt 10\\\underline{\,\,\,-7\,\,-7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+2\,\,\,+2}\\\underline{2y}<\underline{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-3y}<\underline{12}\\{2}\,\,\,\,\,\,\,\,\,\,\,{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-3}\,\,\,\,\,\,\,\,\,\,\,{-3}\\y<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y\gt -4\\y<3\,\,\,\,\textit{or}\,\,\,\,y\gt -4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

The inequality sign is reversed with division by a negative number.

Since y could be less than [latex]3[/latex] or greater than [latex]−4[/latex], y could be any number. Graphing the inequality helps with this interpretation.

Inequality notation: [latex]y<3\text{ or }y> -4[/latex]

Interval notation: [latex]\left(-\infty,\infty\right)[/latex]

Graph:

Even though the graph shows empty dots at [latex]y=3[/latex] and [latex]y=-4[/latex], they are included in the solution.

Solve each inequality separately. Combine the solutions.

[latex]\displaystyle \begin{array}{r}5z-3>18\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-2z-1>15\\\underline{\,\,\,+3\,\,\,+3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,\,+1}\\\underline{5z}>\underline{-15}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-2z}>\underline{16}\\{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-2}\,\,\,\,\,\,\,\,\,\,\,{-2}\\z>-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z<-8\\z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality notation: [latex]\displaystyle z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8[/latex] Interval notation: [latex]\left(-\infty,-8\right)\cup\left(-3,\infty\right)[/latex] Note how we write the intervals with the one containing the most negative solutions first then move to the right on the number line. [latex]z<-8[/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[/latex] has solutions that continue all the way to the right.

Graph: