To graph the inequality [tex]\( y > -x + 2 \)[/tex], we need to break it down into several steps:
1. Graph the boundary line:
- First, plot the boundary line [tex]\( y = -x + 2 \)[/tex].
- This line is in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m = -1 \)[/tex] and the y-intercept [tex]\( b = 2 \)[/tex].
- Plot the y-intercept: Start at the point [tex]\((0, 2)\)[/tex] because when [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex].
- To find another point on the line, use the slope [tex]\( -1 \)[/tex], which means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1 unit.
- From the point [tex]\((0, 2)\)[/tex], move 1 unit to the right to [tex]\((1, 1)\)[/tex].
- Plot another point: Starting from [tex]\((0, 2)\)[/tex], you can also move 1 unit to the left ([tex]\(-1\)[/tex]) and 1 unit up (+1), thus reaching [tex]\((-1, 3)\)[/tex].
2. Draw the boundary line:
- Since the inequality is [tex]\( y > -x + 2 \)[/tex] (strictly greater, not equal), the boundary line [tex]\( y = -x + 2 \)[/tex] will be dashed, indicating that the points on this line are not included in the solution set.
3. Determine the region to shade:
- The inequality [tex]\( y > -x + 2 \)[/tex] indicates that [tex]\( y \)[/tex] is greater than the line [tex]\( y = -x + 2\)[/tex].
- Choose a test point not on the line to see which side of the line to shade. A simple test point is the origin [tex]\((0, 0)\)[/tex].
- Substitute [tex]\((0, 0)\)[/tex] into the inequality: [tex]\( 0 > -0 + 2 \implies 0 > 2 \)[/tex].
- This statement is false, so the region [tex]\((0, 0)\)[/tex] does not satisfy the inequality. Thus, we shade the opposite side of the line (above the dashed line).
4. Interpret the graph:
- The solution set is the region above the dashed line [tex]\( y = -x + 2 \)[/tex].
By following these steps, you can accurately graph the inequality [tex]\( y > -x + 2 \)[/tex] and determine which answer matches the graph. The correct graph will show a dashed line representing [tex]\( y = -x + 2 \)[/tex] and the region above this line shaded.
