which is the graph of linear inequality 2y > x - 2?

Answer:

1. First, rewrite the inequality $2y>x - 2$ in slope - intercept form $y=mx + b$.
   • Divide both sides of the inequality by 2: $y>\frac{1}{2}x-1$.
2. Analyze the boundary line:
   • The equation of the boundary line is $y = \frac{1}{2}x-1$. The slope $m=\frac{1}{2}$ and the y - intercept $b=-1$.
   • Since the inequality is $y>\frac{1}{2}x - 1$ (not $y\geq\frac{1}{2}x-1$), the boundary line is a dashed line.
3. Test a point:
   • A common point to test is the origin $(0,0)$.
   • Substitute $x = 0$ and $y = 0$ into the inequality $y>\frac{1}{2}x-1$. We get $0>\frac{1}{2}(0)-1$, which simplifies to $0>-1$. This is a true statement.
   • Since the origin satisfies the inequality, the region that contains the origin is the solution region.

The graph with a dashed line having a slope of $\frac{1}{2}$ and y - intercept of - 1 and the region above the line and containing the origin is the correct graph.

Explanation:

Step1: Rewrite the inequality

Divide $2y>x - 2$ by 2 to get $y>\frac{1}{2}x-1$.

Step2: Identify the boundary line

The boundary line is $y=\frac{1}{2}x - 1$, dashed for $>$.

Step3: Test a point

Test $(0,0)$ in $y>\frac{1}{2}x-1$, $0>-1$ is true, so region with origin is solution.