Graph the solution:
3x-y-7<0
x+5y+3>=0
This is my thought process in this:
The first equation I can solve for y since we don't have to worry what x is. And I get y>3x-7.
The second one, however, y is being multiplied by 5, so I can't just divide by 5 on both sides because I don't know if the value of y will be positive or negative (and then I'll have to switch the sign).
...So I don't know about the second equation. Help please?
Thank you!
x+5y+3>=0
5 y > -x -3
y > -x/5 -3/5
You divided by a positive number, so there is no reason to reverse the direction of the arrow. Now graph that line and shade above it.
5y ≥ -x - 3
but you are dividing by 5, so you don't have to worry about the inequality sign.
so
y ≥ -x/5 - 3/5
Now in each case, the equation becomes the boundary.
So for the first one, draw a dotted line, and shade in everything above that line.
For the second, draw a dotted line and shade in everything above that line.
To graph the solution to the inequalities 3x - y - 7 < 0 and x + 5y + 3 >= 0, you can follow these steps:
1. Start by graphing the first inequality, 3x - y - 7 < 0.
- Turn the inequality into an equation by replacing the less than symbol (<) with an equal sign (=). So, it becomes 3x - y - 7 = 0.
- Solve this equation for y to get y = 3x - 7. This is the equation of a straight line.
- To graph this line, start by choosing some x values and substituting them into the equation to find the corresponding y values.
- Plot these points on a coordinate grid and connect them to form a line.
- Lastly, since the inequality is less than, you need to shade the region below the line.
2. Now, let's look at the second inequality, x + 5y + 3 >= 0.
- Turn the inequality into an equation by replacing the greater than or equal to symbol (>=) with an equal sign (=). So, it becomes x + 5y + 3 = 0.
- Solve this equation for y to get y = (-x - 3)/5. This is also the equation of a straight line.
- However, in this case, you noticed that y is being multiplied by 5, so you cannot simply divide by 5 on both sides to isolate y due to the potential sign change.
- To graph this line, you can use a different approach. Use x-intercepts and y-intercepts to plot the line:
- Set x = 0 to find the y-intercept. You get y = (-0 - 3)/5 = -3/5.
- Set y = 0 to find the x-intercept. You get x + 5(0) + 3 = 0, which gives x = -3.
- Plot these two points on a coordinate grid and connect them to form a line.
- Since the inequality is greater than or equal to, you need to shade the region above the line.
In summary, graph the lines y = 3x - 7 and y = (-x - 3)/5 on a coordinate grid, and shade the region below the first line and above the second line to represent the solution to the given system of inequalities.