y>−x+2
y≤3x−2
Which graph represents the system of inequalities?
To graph the system of inequalities y > -x + 2 and y ≤ 3x - 2, we will start by graphing each inequality separately and then identify the overlapping region.
First, let's graph the inequality y > -x + 2.
Step 1: Replace the inequality symbol with an equal sign to graph the boundary line.
Thus, the equation becomes y = -x + 2.
Step 2: Graph the line y = -x + 2 on the coordinate plane.
To do this, we locate the y-intercept at (0,2) and find another point by using the slope of -1 (since the coefficient of x is -1). We can use the point (1,1) or (-1,3) as the second point, and then draw a straight line passing through these two points.
Step 3: Since the inequality symbol is y > -x + 2, the boundary line should be drawn as a dashed line since y is greater than -x + 2, not inclusive.
Next, let's graph the inequality y ≤ 3x - 2.
Step 1: Replace the inequality symbol with an equal sign to graph the boundary line.
Thus, the equation becomes y = 3x - 2.
Step 2: Graph the line y = 3x - 2 on the coordinate plane.
To do this, we locate the y-intercept at (0,-2) and find another point by using the slope of 3. We can use the point (1,1) or (-1,-5) as the second point, and then draw a straight line passing through these two points.
Step 3: Since the inequality symbol is y ≤ 3x - 2, the boundary line should be drawn as a solid line since y is less than or equal to 3x - 2.
Finally, we need to determine the overlapping region of the two inequalities.
Since the first inequality, y > -x + 2, has a dashed boundary line, we should shade the region above the line.
Since the second inequality, y ≤ 3x - 2, has a solid boundary line, we should shade the region below or on the line.
The overlapping region represents the solution to the system of inequalities.
Without a visual display, it is challenging to provide a specific graph, but you can refer to the instructions above to sketch the correct graph.
To graph the system of inequalities y > -x + 2 and y ≤ 3x - 2, we can follow these steps:
1. Start by graphing the first inequality, y > -x + 2.
- To graph this inequality, we need to first draw the boundary line y = -x + 2.
- The boundary line is the line that represents the equation without the inequality sign.
- To do this, we can plot two points on the line, or find the x and y intercepts.
- Let's find the x and y intercepts by setting x = 0 and y = 0, respectively:
- When x = 0, y = -0 + 2 = 2, giving us the point (0, 2).
- When y = 0, 0 = -x + 2, which implies -x = -2, so x = 2. Therefore, the point is (2, 0).
- Plot these points and draw a line passing through them.
- Since the inequality is y > -x + 2, we need to shade the region above the boundary line (not including the line) because y is greater than the equation.
2. Next, graph the second inequality, y ≤ 3x - 2.
- Start by drawing the boundary line y = 3x - 2, following the same process as before.
- Find the x and y intercepts by substituting x = 0 and y = 0 into the equation:
- When x = 0, y = -2, giving us the point (0, -2).
- When y = 0, 0 = 3x - 2, so 3x = 2, and x = 2/3. This gives us the point (2/3, 0).
- Plot these points and draw a line passing through them.
- The inequality is y ≤ 3x - 2, so we should shade the region below the boundary line, including the line, since it is less than or equal to.
3. Now, look at the shaded regions for each inequality.
- The solution to the system of inequalities is the overlapping region between the two shaded areas.
- Sketch this region on your graph by shading the area where the two shaded regions overlap.
The graph representing the system of inequalities should have the region shaded where the shaded areas of both inequalities overlap.