In which quadrants do solutions for the inequality y ≤ 2/7x +1 exist? ~ I, III, and IV~ I, II, and III~ I and IV~ All four quadrants

All four quadrantsExplanation:We have the following inequality:[tex]y \leq \frac{2}{7}x+1[/tex]So the first step we need to perform is to plot the line:[tex]y = \frac{2}{7}x+1[/tex][tex]If \ x=0 \\ \\ y=\frac{2}{7}(0)+1 \\ \\ y=1 \\ \\ \\ If \ y=0: \\ \\ 0=\frac{2}{7}(x)+1 \\ \\ x=-\frac{7}{2}=-3.5[/tex]So the line passes through the points:[tex](0,1) \ and \ (-3.5,0)[/tex]To find the shaded region, let us take a point, namely, the origin and test it in the inequality:[tex]y \leq \frac{2}{7}x+1 \\ \\ 0\leq \frac{2}{7}(0)+1 \\ \\ 0\leq 1 \ True![/tex]Since this is true, then the shaded region includes this point. This is shown below and as you can see the solutions exist in all four quadrants.Learn more:Inequalities: