The graphed inequality is:
[tex]3y+18>5x[/tex]
Write the equation for y:
-subtract 18 from both sides
[tex]\begin{gathered} 3y+18-18\Rightarrow5x-18 \\ 3y>5x-18 \end{gathered}[/tex]
-Divide both sides by 3
[tex]\begin{gathered} \frac{3y}{3}>\frac{5x}{3}-\frac{18}{3} \\ y>\frac{5}{3}x-6 \end{gathered}[/tex]
The symbol of the inequality ">" indicates that its range corresponds to the area above the line determined by the expression but without including the values over the line. To represent this graphically, the line should be drawn with as a dotted line.
The y-intercept of the line can be determined by replacing the inequality with x=0:
[tex]\begin{gathered} y>\frac{5}{3}x-6 \\ y>\frac{5}{3}\cdot0-6 \\ y>-6 \end{gathered}[/tex]
The y-intercept is (0,-6)
To determine the x-intercept you have to zero the inequality and calculate the corresponding value of x:
[tex]\begin{gathered} 0>\frac{5}{3}x-6 \\ 0+6>\frac{5}{3}x-6+6 \\ 6>\frac{5}{3}x \\ 6\cdot\frac{3}{5}>(\frac{5}{3}\cdot\frac{3}{5})x \\ \frac{18}{5}>x \\ 3.6>x \end{gathered}[/tex]
The x-intercept is (3.6,0)
Using both points you can determine which graph correspond to the given inequality:
The graph that corresponds to the inequality is option D.
