Can you please look over these and let me know if they are right?
1/4(5x-1/2)-5/3<1/3
5/4x-1/8-5/3<1/3
24(5/4x-1/8-5/3)<24(1/3)
30x-3-40<8
30x-43+43<8+43
30x<51
x<51/30
{x|x<51/30}
31-(2x+2)< or equal to 4(x+1)+x
31-2x-2< equal to 4x+4+x
29-2x<equal to 5x+4
-2x<equal to 5x-25
-7x<equal to -25
{x|x>equal to 3.6}
Use elimination method: 4x+5y=-12 and 7x-5y=-76 I got x=-8 and y=4
elimination: 0.3x-0.2y=4 and 0.4x+0.5y=1 I got the solution of (220/23,-130/23)
correct on all.
Let's check the answers step by step:
1. 1/4(5x-1/2)-5/3<1/3
To solve this inequality, we can follow these steps:
- Distribute the 1/4 to the terms inside the parentheses: (5/4)x - 1/8 - 5/3 < 1/3.
- Combine the constant terms: (5/4)x - 1/8 - 15/8 < 1/3.
- Combine the fractions: (5/4)x - 16/8 < 1/3.
- Simplify: (5/4)x - 2 < 1/3.
- Add 2 to both sides: (5/4)x < 1/3 + 2.
- Convert the mixed fraction to an improper fraction: (5/4)x < 7/3.
- Multiply both sides by the reciprocal of (5/4), which is 4/5: x < (7/3) * (4/5).
- Simplify: x < 28/15.
Therefore, the correct solution is x < 28/15.
2. 31-(2x+2) ≤ 4(x+1)+x
Let's solve this inequality:
- Distribute the multiplication: 31 - 2x - 2 ≤ 4x + 4 + x.
- Simplify: 29 - 2x ≤ 5x + 4.
- Subtract 5x from both sides: 29 - 7x ≤ 4.
- Subtract 29 from both sides: -7x ≤ -25.
- Divide by -7 (since we are dividing by a negative number, we need to flip the direction of the inequality): x ≥ -25/-7.
- Simplify: x ≥ 25/7.
Therefore, the correct solution is x ≥ 25/7.
3. Using the elimination method to solve 4x + 5y = -12 and 7x - 5y = -76:
First, let's simplify the equations:
4x + 5y = -12 (Equation 1)
7x - 5y = -76 (Equation 2)
To eliminate the variable y, we can add these two equations together:
(Equation 1) + (Equation 2): (4x + 5y) + (7x - 5y) = -12 + (-76).
Simplifying: 4x + 7x = -12 - 76.
Combining like terms: 11x = -88.
Divide both sides by 11 to solve for x: x = -88/11.
Simplify: x = -8.
Now substitute the value of x back into one of the original equations to solve for y:
4(-8) + 5y = -12.
-32 + 5y = -12.
Add 32 to both sides: 5y = -12 + 32.
Combine like terms: 5y = 20.
Divide both sides by 5 to solve for y: y = 20/5.
Simplify: y = 4.
Therefore, the solution to the system of equations is x = -8 and y = 4.
4. Using the elimination method to solve 0.3x - 0.2y = 4 and 0.4x + 0.5y = 1:
Let's eliminate the variable y by multiplying both equations:
(10)(0.3x - 0.2y) = (10)(4)
and
(5)(0.4x + 0.5y) = (5)(1).
These become:
3x - 2y = 40 (Equation 3)
2x + 2.5y = 5 (Equation 4).
Now let's add Equation 3 + Equation 4:
(3x - 2y) + (2x + 2.5y) = 40 + 5.
Simplifying: 3x + 2x - 2y + 2.5y = 45.
Combining like terms: 5x + 0.5y = 45.
To eliminate the decimal, we can multiply both sides by 2:
(2)(5x + 0.5y) = (2)(45).
10x + y = 90.
Now we have the following two equations:
10x + y = 90 (Equation 5)
2x + 2.5y = 5 (Equation 4).
To eliminate the variable y, we can multiply Equation 5 by -2 and Equation 4 by 1:
(-2)(10x + y) = (-2)(90)
and
(1)(2x + 2.5y) = (1)(5).
These become:
-20x - 2y = -180 (Equation 6)
2x + 2.5y = 5 (Equation 4).
Finally, let's add Equation 6 + Equation 4:
(-20x - 2y) + (2x + 2.5y) = -180 + 5.
Simplifying: -20x + 2x - 2y + 2.5y = -175.
Combining like terms: -18x + 0.5y = -175.
To eliminate the decimal, we can multiply both sides by 2:
(2)(-18x + 0.5y) = (2)(-175).
-36x + y = -350.
Now we have the following two equations:
-36x + y = -350 (Equation 7)
-18x + 0.5y = -175 (Equation 8).
To eliminate the variable y, we can multiply Equation 7 by 0.5 and Equation 8 by 2:
(0.5)(-36x + y) = (0.5)(-350)
and
(2)(-18x + 0.5y) = (2)(-175).
These become:
-18x + 0.5y = -175 (Equation 8)
-36x + 0.5y = -175 (Equation 9).
Notice that Equation 8 and Equation 9 are the same, which means they represent the same line. Therefore, there are infinitely many solutions to this system of equations.
In conclusion, the correct solution for the system of equations is an infinite number of points in the form (x, -18x + 0.5y) where x and y can take any values.