x+2y<4
x-y<5
*6x-4
*36-(50-15x)=10x
To solve the given system of inequalities:
1. Simplify the two inequalities:
a) x + 2y < 4
b) x - y < 5
2. Next, we need to determine the values of x and y that satisfy both inequalities. This can be done by graphing the inequalities on a coordinate plane and finding the region where the shaded regions overlap. However, let's solve the system algebraically:
3. Solve the first inequality:
a) x + 2y < 4
Subtract x from both sides:
2y < 4 - x
Divide both sides by 2:
y < (4 - x) / 2
Rewrite the inequality:
y < 2 - x/2 ... (Equation 1)
4. Solve the second inequality:
b) x - y < 5
Add y to both sides:
x < 5 + y
Rewrite the inequality:
x - y < 5 ... (Equation 2)
5. Now, we have the two inequalities:
a) y < 2 - x/2
b) x - y < 5
6. We can plot these inequalities on a graph, but let's first simplify the two equations at the bottom:
Multiply equation 1 by (-1):
-y > -2 + x/2
Simplify equation 2 further:
x - y < 5
Add y to both sides:
x < 5 + y
7. Rearrange equation 2:
y > x - 5 ... (Equation 3)
8. Now we have the following simplified inequalities:
a) -y > -2 + x/2 ... (Equation 4)
b) y > x - 5 ... (Equation 5)
9. Graphing these inequalities would intersect the regions where the shaded areas overlap, but we will continue solving algebraically:
10. Combine equations 4 and 5 by multiplying equation 5 by -1:
-y > -x + 5
11. Combine the two inequalities:
-y > -2 + x/2 ... (Equation 4)
-y > -x + 5 ... (Equation 6)
12. Multiply equation 6 by (-1) to switch the inequality sign:
y < x - 5
13. Now we have:
a) -y > -2 + x/2
b) y < x - 5
14. Graph or plot these two inequalities on a graph and find the overlapping region. This overlapping region represents the solutions to the given system of inequalities.
Now, let's move on to the equations:
1. 6x - 4:
This is a simple equation with no variables. To solve it, you would subtract 4 from 6x:
6x - 4 = 0
6x = 4
x = 4/6
Simplify the fraction if needed.
2. 36 - (50 - 15x) = 10x:
Start by simplifying the parentheses within the equation:
36 - 50 + 15x = 10x
Combine like terms:
-14 + 15x = 10x
Move both the x terms to one side and the constant terms to the other side:
15x - 10x = 14
5x = 14
Divide both sides by 5:
x = 14/5
Simplify the fraction if needed.
By following the steps outlined, you can solve the system of inequalities and solve the given equations.