Solve each system of inequality by graphing
17)y>-x+5 y<3x-4
Can someone help me on this i just don't understand it that well
I will assume there is a comma between them in this way:
y>-x+5 , y<3x-4
First one:
y > -x + 5 make a sketch of y = -x + 5 and shade in the region above but excluding the line
We do that by drawing the line as a dotted line
Second one:
y < 3x - 4
again sketch y = 3x - 4, and shade in the region below but excluding the line itself.
your solution is the region shaded by both regions, but excluding the boundary lines
btw, the two boundary lines intersect at
3x-4 = -x+ 5
4x = 9
x = 9/4 , y = -9/4+5 = 7/4 , that is, at (9/4, 7/4)
Um yes i forgot about the commas anyways thank you so much now i understand a little bit better thank you :)
Certainly! I can help you with that. To solve a system of inequalities by graphing, we will first graph each inequality separately and then find the overlapping region, known as the solution region.
Let's start by graphing the first inequality, y > -x + 5. To do this, we will begin by graphing the boundary line, which is the line corresponding to the equation y = -x + 5.
To graph this line, we can start by finding its intercepts. The y-intercept is 5, so we plot the point (0, 5) on the y-axis. The x-intercept can be found by setting y = 0 in the equation and solving for x:
0 = -x + 5
x = 5. Therefore, we plot the point (5, 0) on the x-axis.
Now, draw a dotted line connecting these two points. Since the inequality is y > -x + 5, the line should be dashed, indicating that it is not included in the solution.
Next, to determine which side of the line to shade, pick a point not on the line and substitute its coordinates into the inequality. For example, we can use the point (0, 0). Substituting these values into the inequality, we get:
0 > -0 + 5
0 > 5
Since this is not true, we do not shade the side of the line containing the point (0, 0). Instead, shade the opposite side.
Now let's graph the second inequality, y < 3x - 4. Again, we begin by graphing its boundary line, which is the line corresponding to the equation y = 3x - 4.
To graph this line, find its intercepts. The y-intercept is -4, so plot the point (0, -4) on the y-axis. The x-intercept can be found by setting y = 0 in the equation and solving for x:
0 = 3x - 4
4 = 3x
x = 4/3. Therefore, plot the point (4/3, 0) on the x-axis.
Draw a dashed line connecting these two points since the inequality is y < 3x - 4.
To determine which side of the line to shade, pick a test point not on the line, such as (0, 0). Substitute these values into the inequality:
0 < 3(0) - 4
0 < -4
This is true, so shade the side of the line containing the point (0, 0).
Now that we have graphed both inequalities, we can examine their overlapping region, which represents the solution to the system. The solution region is the region that satisfies both inequalities.
In this case, the overlapping region is the shaded area between the two dashed lines.
To summarize, to solve the system of inequalities y > -x + 5 and y < 3x - 4 by graphing:
1. Graph the first inequality y > -x + 5 by drawing a dashed line and shading the opposite side.
2. Graph the second inequality y < 3x - 4 by drawing a dashed line and shading the side containing the origin.
3. The solution is the overlapping region between the two dashed lines.
I hope this explanation helps you understand how to solve a system of inequalities by graphing!