Solve the inequalities by graphing
y<=4x-1
y>3x-1
To solve these inequalities by graphing, we will plot the graphs of the two inequalities on the coordinate plane and identify the shaded regions that satisfy each inequality.
First, let's solve the first inequality, y <= 4x - 1:
Step 1: Start by graphing the equation y = 4x - 1, which is the boundary line for this inequality. To plot the line, we can find two points on the line and draw a straight line passing through them.
For example, if we set x = 0, we get y = -1. So, one point on the line is (0, -1).
If we set x = 1, we get y = 3. So, another point on the line is (1, 3).
Plot these two points on the coordinate plane and draw a line passing through them.
Step 2: To determine which side of the line to shade, we can choose a point not on the line and check if it satisfies the inequality. The origin (0, 0) is convenient to use.
For (0, 0):
y <= 4x - 1
0 <= 4(0) - 1
0 <= -1
Since 0 is not less than or equal to -1, the side of the line containing the origin is not in the shaded region.
Step 3: Therefore, we shade the other side of the line to represent the region that satisfies y <= 4x - 1. This shaded region indicates all the possible solutions for the first inequality.
Next, let's solve the second inequality, y > 3x - 1:
Step 1: Graph the equation y = 3x - 1, which is the boundary line for this inequality. Following the same steps as before, plot two points on the line and draw a line passing through them.
If we set x = 0, we get y = -1. So, one point on the line is (0, -1).
If we set x = 1, we get y = 2. So, another point on the line is (1, 2).
Plot these points and draw a line.
Step 2: Choose a point not on the line to determine the shaded region. Again, we can use the origin (0, 0).
For (0, 0):
y > 3x - 1
0 > 3(0) - 1
0 > -1
Since 0 is greater than -1, the origin is in the shaded region.
Step 3: Shade the side of the line that includes the origin to represent the region that satisfies y > 3x - 1.
Once you have both shaded regions, the overlapping region will represent the solutions that satisfy both inequalities.
To solve the inequalities by graphing, we need to plot the equations on a coordinate plane and shade the regions that satisfy the given inequalities.
First, for the inequality y <= 4x - 1:
1. Plot the line y = 4x - 1. To do this, identify two points on the line by assigning values to x and solving for y. For example, if we set x = 0, we get y = -1. If we set x = 2, we get y = 7. This gives us two points (0, -1) and (2, 7). Plot these points and draw a straight line passing through them.
2. Since the inequality is "y <= 4x - 1", we need to shade the region below the line. You can do this by drawing a dashed line (since the inequality includes equality) and shading the area below it.
Next, for the inequality y > 3x - 1:
1. Plot the line y = 3x - 1. Use the same process as before to find points on the line. For example, if we set x = 0, we get y = -1. If we set x = 2, we get y = 5. Plot these points and draw a straight line passing through them.
2. Since the inequality is "y > 3x - 1", we need to shade the region above the line. You can do this by drawing a dashed line (since the inequality does not include equality) and shading the area above it.
The shaded region that satisfies both inequalities is the region where the shaded areas from each inequality overlap.
Draw the lines
y = 4x-1
y = 3x-1
Shade everything below the 1st, and above the 2nd.
The intersection of the two shaded areas is the solution set