System Inequalities 82Bb76

1. **State the problem:** We have a system of two inequalities represented by shaded regions on a graph. We want to write the system and determine which points satisfy both inequalities. 2. **Write the boundary lines:** - The first boundary line passes through points $(-3,3)$ and $(3,-3)$. The slope is $m=\frac{-3-3}{3-(-3)}=\frac{-6}{6}=-1$. - Using point-slope form with point $(-3,3)$: $$y-3=-1(x+3)\implies y-3=-x-3\implies y=-x$$ 3. **Inequality for the first line:** - The shading is above the line $y=-x$, so the inequality is: $$y \geq -x$$ 4. **Second boundary line:** - Passes through $(-1,3)$ and $(3,-1)$. - Slope: $$m=\frac{-1-3}{3-(-1)}=\frac{-4}{4}=-1$$ - Equation using point $(-1,3)$: $$y-3=-1(x+1)\implies y-3=-x-1\implies y=-x+2$$ 5. **Inequality for the second line:** - Shading is below the line $y=-x+2$, so: $$y \leq -x+2$$ 6. **System of inequalities:** $$\begin{cases} y \geq -x \\ y \leq -x+2 \end{cases}$$ 7. **Check each point to see if it satisfies both inequalities:** - Point $C(-2,3)$: - Check $y \geq -x$: $3 \geq -(-2) \Rightarrow 3 \geq 2$ (True) - Check $y \leq -x+2$: $3 \leq -(-2)+2 \Rightarrow 3 \leq 4$ (True) - So $C$ satisfies both. - Point $F(-1.5,2)$: - $2 \geq -(-1.5) \Rightarrow 2 \geq 1.5$ (True) - $2 \leq -(-1.5)+2 \Rightarrow 2 \leq 3.5$ (True) - $F$ satisfies both. - Point $G(0,1.5)$: - $1.5 \geq -0 \Rightarrow 1.5 \geq 0$ (True) - $1.5 \leq -0+2 \Rightarrow 1.5 \leq 2$ (True) - $G$ satisfies both. - Point $A(1,1.5)$: - $1.5 \geq -1 \Rightarrow 1.5 \geq -1$ (True) - $1.5 \leq -1+2 \Rightarrow 1.5 \leq 1$ (False) - $A$ does not satisfy both. - Point $B(1,0.5)$: - $0.5 \geq -1 \Rightarrow 0.5 \geq -1$ (True) - $0.5 \leq -1+2 \Rightarrow 0.5 \leq 1$ (True) - $B$ satisfies both. - Point $D(-1,-1.5)$: - $-1.5 \geq -(-1) \Rightarrow -1.5 \geq 1$ (False) - $D$ does not satisfy both. - Point $E(1.5,-2)$: - $-2 \geq -1.5 \Rightarrow -2 \geq -1.5$ (False) - $E$ does not satisfy both. - Point $H(3,-2.5)$: - $-2.5 \geq -3 \Rightarrow -2.5 \geq -3$ (True) - $-2.5 \leq -3+2 \Rightarrow -2.5 \leq -1$ (True) - $H$ satisfies both. **Final answer:** Points $C$, $F$, $G$, $B$, and $H$ satisfy the system.