Determine the solution set of 2x-3y<6
4x+4y<8
do I graph this?
Take a look here:
http://www.wolframalpha.com/input/?i=solve+2x-3y%3C6%2C+4x%2B4y%3C8+
To determine the solution set of the system of inequalities 2x - 3y < 6 and 4x + 4y < 8, you have a few options. Graphing is one method, but there are also algebraic methods that can be used. Let's take a look at both approaches:
1. Graphical Method:
To graph the system of inequalities, we'll convert them into their corresponding equations and then plot the corresponding lines on a coordinate plane.
For the first inequality, 2x - 3y < 6, we'll consider the equation 2x - 3y = 6.
Rearranging it, we get: 3y = 2x - 6, or y = (2/3)x - 2.
For the second inequality, 4x + 4y < 8, we'll consider the equation 4x + 4y = 8.
Rearranging it, we get: 4y = -4x + 8, or y = (-1)x + 2.
Plotting these lines on a Cartesian coordinate plane will give us a better understanding of the solution set. Once the lines are graphed, you'll need to identify the region that satisfies both inequalities. This region represents the solution set.
2. Algebraic Method:
For a quicker algebraic solution, you can solve each inequality for y and then compare the resulting inequalities to determine the overlapping region.
Starting with the first inequality, 2x - 3y < 6:
Add 3y to both sides: 2x < 3y + 6.
Divide both sides by 3: (2/3)x < y + 2.
Now, rearrange the terms to get y isolated: y > (2/3)x - 2.
Now, let's work on the second inequality, 4x + 4y < 8:
Subtract 4x from both sides: 4y < -4x + 8.
Divide both sides by 4: y < (-1)x + 2.
Now we have two inequalities: y > (2/3)x - 2 and y < (-1)x + 2.
The solution set will be the region that satisfies both inequalities simultaneously. You can see this by shading the region on a graph or by analyzing the relationship between the two inequalities.
To summarize, while graphing is an option to determine the solution set of the given system of inequalities, an algebraic approach is also feasible by solving for y in each inequality and comparing the resulting inequalities to establish the overlapping region.