identifying the graph of an inequality\nwhich is the graph of 2x - 4y > 6?

Answer:

First, rewrite the inequality $2x - 4y>6$ in slope - intercept form $y=mx + b$.

1. Start by isolating $y$:
   • Subtract $2x$ from both sides of the inequality: $-4y>-2x + 6$.
   • Divide each term by $-4$. Remember, when dividing or multiplying an inequality by a negative number, the direction of the inequality sign changes. So we get $y<\frac{1}{2}x-\frac{3}{2}$.
2. Analyze the key features of the line $y = \frac{1}{2}x-\frac{3}{2}$:
   • The slope $m=\frac{1}{2}$, which means for every increase of 1 in the $x$ - value, the $y$ - value increases by $\frac{1}{2}$.
   • The $y$ - intercept $b =-\frac{3}{2}=-1.5$.
3. Analyze the inequality $y<\frac{1}{2}x-\frac{3}{2}$:
   • Since the inequality is $y<\frac{1}{2}x-\frac{3}{2}$, the line $y = \frac{1}{2}x-\frac{3}{2}$ will be a dashed line (because the inequality is strict, i.e., $y$ is strictly less than, not less than or equal to).
   • To find which side of the line to shade, we can test a point. Let's test the point $(0,0)$.
     • Substitute $x = 0$ and $y = 0$ into the original inequality $2x-4y>6$. We get $2(0)-4(0)=0$, and $0\not>6$. So, we shade the side of the line that does not contain the point $(0,0)$.

The graph of the inequality $y<\frac{1}{2}x-\frac{3}{2}$ has a dashed line with a slope of $\frac{1}{2}$ and a $y$ - intercept of $-1.5$, and the shading is below the line.

Explanation:

Step1: Rewrite in slope - intercept form

Subtract $2x$ from both sides: $-4y>-2x + 6$.

Step2: Solve for $y$

Divide by $-4$ and reverse the inequality: $y<\frac{1}{2}x-\frac{3}{2}$.

Step3: Identify slope and y - intercept

Slope $m = \frac{1}{2}$, $y$ - intercept $b=-\frac{3}{2}$.

Step4: Determine line type

Since $y<\frac{1}{2}x-\frac{3}{2}$, the line is dashed.

Step5: Find the shaded region

Test the point $(0,0)$ in the original inequality. Since $0\not>6$, shade the side not containing $(0,0)$.