Solve each system of inequalities by graphing. 6. ) yleqslant x-4 ygt vert x-6vert 7. ) 2x+yleqslant 3 ygt vert x+3vert -2

To solve each system of inequalities by graphing, we need to graph each inequality on the same coordinate plane and find the region where all inequalities overlap.<br /><br />### Problem 6: <br />\[<br />\{ \begin{matrix} y \leqslant x - 4 \\ y > |x - 6| \end{matrix}<br />\]<br /><br />1. **Graph \( y \leqslant x - 4 \)**:<br /> - This is a line with slope 1 and y-intercept -4.<br /> - The boundary line is \( y = x - 4 \).<br /> - Shade the region below this line (including the line itself).<br /><br />2. **Graph \( y > |x - 6| \)**:<br /> - This represents two lines: \( y > x - 6 \) and \( y > -(x - 6) \).<br /> - For \( y > x - 6 \), the boundary line is \( y = x - 6 \). Shade above this line.<br /> - For \( y > -(x - 6) \), rewrite as \( y > -x + 6 \). The boundary line is \( y = -x + 6 \). Shade above this line.<br /><br />The solution region is where the shaded areas of both inequalities overlap.<br /><br />### Problem 7:<br />\[<br />\{ \begin{matrix} 2x + y \leqslant 3 \\ y > |x + 3| - 2 \end{matrix}<br />\]<br /><br />1. **Graph \( 2x + y \leqslant 3 \)**:<br /> - This is a line with slope -2 and y-intercept 3.<br /> - The boundary line is \( y = -2x + 3 \).<br /> - Shade the region below this line (including the line itself).<br /><br />2. **Graph \( y > |x + 3| - 2 \)**:<br /> - This represents two lines: \( y > x + 3 - 2 \) and \( y > -(x + 3) - 2 \).<br /> - Simplify to \( y > x + 1 \) and \( y > -x - 5 \).<br /> - For \( y > x + 1 \), the boundary line is \( y = x + 1 \). Shade above this line.<br /> - For \( y > -x - 5 \), the boundary line is \( y = -x - 5 \). Shade above this line.<br /><br />The solution region is where the shaded areas of both inequalities overlap.<br /><br />By graphing these inequalities on the same coordinate plane, you can visually identify the regions that satisfy all conditions in each system.