Question

graph the solution set of the following system of inequalities.
4x + 8y ≤ 8
3x + y ≤ 9
use the graphing tool to graph the system of inequalities.
click to enlarge graph

Explanation:

Step1: Rewrite first inequality in slope - intercept form

Starting with $4x + 8y\leq8$, we solve for $y$. First, subtract $4x$ from both sides: $8y\leq - 4x + 8$. Then divide by 8: $y\leq-\frac{1}{2}x + 1$. The boundary line is $y =-\frac{1}{2}x + 1$, which has a y - intercept of 1 and a slope of $-\frac{1}{2}$. Since the inequality is $\leq$, the region below the line (including the line itself) is part of the solution.

Step2: Rewrite second inequality in slope - intercept form

Starting with $3x + y\leq9$, we solve for $y$. Subtract $3x$ from both sides: $y\leq - 3x+9$. The boundary line is $y=-3x + 9$, which has a y - intercept of 9 and a slope of - 3. Since the inequality is $\leq$, the region below the line (including the line itself) is part of the solution.

Step3: Find the intersection point

Set $-\frac{1}{2}x + 1=-3x + 9$. Add $3x$ to both sides: $-\frac{1}{2}x+3x + 1=-3x+3x + 9$, which simplifies to $\frac{-1 + 6}{2}x+1 = 9$, or $\frac{5}{2}x+1 = 9$. Subtract 1 from both sides: $\frac{5}{2}x=8$. Multiply both sides by $\frac{2}{5}$: $x=\frac{16}{5}=3.2$. Substitute $x = 3.2$ into $y=-3x + 9$, we get $y=-3\times3.2 + 9=-9.6 + 9=-0.6$. The intersection point of the two boundary lines is $(3.2,-0.6)$.

Answer:

Graph the line $y =-\frac{1}{2}x + 1$ as a solid line and shade the region below it. Graph the line $y=-3x + 9$ as a solid line and shade the region below it. The solution set of the system of inequalities is the intersection of the two shaded regions (including the boundary lines).