1. x>15-2x / -3 Solve the linear inequality. Write the solution in set builder notation.
So, -3x>15-2x, x<-15..?
And how do I write it in set builder notation?
2. y+x=9 and 3y=2x + 8. How do I solve this system of equations?
3. Find the maximum y- value of the following quadratic function: f(x)= -2x^2 + 7x + 1
4. Solve the following equation: 4x/3 + 1/2 = 5y/6
I'm not even sure what this question is asking...
5. Find the solution set to the equation below: 2*absolute value of x-100 = 50 +absolute value of x-100.
x = 150?
x > (15-2x)/-3
looks like you multiplied both sides by -3, but did not reverse the inequality sign.
should have been:
-3x < 15 - 2x
-x < 15
x> -15
{x | x > -15 }
2. I would use substitution.
from the first: y =9-x
sub into the 2nd:
3(9-x) = 2x+8
27 - 3x = 2x + 8
-5x = -19
x = 19/5 = 3.8
y = 9-3.8 = 5.2
3. Find the maximum y- value of the following quadratic function: f(x)= -2x^2 + 7x + 1
The maximum value of the y value is obtained from the vertex
the x of the vertex is -b/(2a) = -7/-4 = 7/4
y = -2(49/16) + 7(7/4) + 1 = 57/8
4.
4x/3 + 1/2 = 5y/6
multiply each term by 6
8x + 3 = 5y
8x + 5y + 3 = 0
----> the equation of a straight line, with slope of -5/3
"solve" is not the correct instruction for this equation.
5.
2|x-100| = 50 + |x-100|
2(x-100) = 50 + |x-100| OR -2(x-100) = 50 + |x-100|
case1 :
2(x-100) = 50 + |x-100|
2x - 200 = 50 + |x-100|
2x - 250 = |x-100|
x-100 = 2x-250 OR x-100 = -2x + 250
-x = -150 OR 3x = 350
x = 150 OR x = 350/3
case2:
-2(x-100) = 50 + |x-100|
-2x + 100-50 = |x-100|
-2x + 50 = |x-100|
x-100 = -2x + 50 OR x-100 = 2x - 50
3x = 150 OR -x = 50
x = 50 or x = -50
x = ±50 , 150 , 350/3
check my arithmetic on the last one
1. To solve the linear inequality x > (15 - 2x) / -3, we can begin by multiplying both sides of the inequality by -3 to remove the fraction. However, we need to be careful when multiplying or dividing by a negative number since it will change the direction of the inequality sign.
Multiplying both sides by -3, we get:
-3x > 15 - 2x
Next, we can simplify by adding 2x to both sides:
-3x + 2x > 15 - 2x + 2x
-x > 15
To isolate x, we can multiply both sides by -1, keeping in mind to flip the inequality sign:
-x*(-1) > 15*(-1)
x < -15
So, the solution to the linear inequality x > (15 - 2x) / -3 is x < -15.
To write the solution in set builder notation, we can express it as:
{x | x < -15}
2. To solve the system of equations y + x = 9 and 3y = 2x + 8, we can use the method of substitution.
First, let's solve the first equation for y:
y = 9 - x
Substitute this expression for y into the second equation:
3(9 - x) = 2x + 8
Simplify and solve for x:
27 - 3x = 2x + 8
5x = 19
x = 19 / 5
Substitute this value of x back into the first equation to solve for y:
y + 19/5 = 9
y = 9 - 19/5
y = 36/5 - 19/5
y = 17/5
So, the solution to the system of equations y + x = 9 and 3y = 2x + 8 is x = 19/5 and y = 17/5.
3. To find the maximum y-value of the quadratic function f(x) = -2x^2 + 7x + 1, we can use the vertex formula. The vertex formula for a quadratic function in the form f(x) = ax^2 + bx + c is given by x = -b / 2a.
In this case, a = -2, b = 7, and c = 1. Plugging these values into the vertex formula, we have:
x = -7 / (2 * -2)
x = -7 / -4
x = 7/4
To find the corresponding y-value, substitute this x-value back into the quadratic function:
f(7/4) = -2(7/4)^2 + 7(7/4) + 1
f(7/4) = -2(49/16) + 49/4 + 1
f(7/4) = -98/16 + 196/16 + 1
f(7/4) = 99/16
Therefore, the maximum y-value of the quadratic function f(x) = -2x^2 + 7x + 1 is 99/16.
4. To solve the equation 4x/3 + 1/2 = 5y/6, we can begin by eliminating the fractions. Multiplying both sides of the equation by the least common multiple (LCM) of the denominators can achieve this.
The LCM of 3 and 2 is 6, so multiply both sides of the equation by 6:
6 * (4x/3) + 6 * (1/2) = 6 * (5y/6)
Simplifying, we have:
8x + 3 = 5y
To solve for y, we can isolate it by subtracting 8x from both sides:
8x + 3 - 8x = 5y - 8x
3 = 5y - 8x
Finally, we can rewrite the equation in the form y = mx + b:
5y = 8x + 3
y = (8x + 3)/5
So, the solution to the equation 4x/3 + 1/2 = 5y/6 is y = (8x + 3)/5.
5. To find the solution set to the equation 2 * |x - 100| = 50 + |x - 100|, we can consider two cases: when (x - 100) is positive and when (x - 100) is negative.
Case 1: (x - 100) is positive
In this case, the absolute value |x - 100| simplifies to (x - 100). Plugging this into the equation, we have:
2(x - 100) = 50 + (x - 100)
Simplifying, we get:
2x - 200 = 50 + x - 100
x = -50
Case 2: (x - 100) is negative
In this case, the absolute value |x - 100| simplifies to -(x - 100). Plugging this into the equation, we have:
2(-(x - 100)) = 50 + (-(x - 100))
Simplifying, we get:
-2x + 200 = 50 - x + 100
x = 150
Therefore, the solution set to the equation 2 * |x - 100| = 50 + |x - 100| is {x | x = -50 or x = 150}.
1. To solve the inequality x > (15 - 2x) / -3, we need to isolate x on one side of the inequality sign.
First, we can multiply both sides of the inequality by -3 to get rid of the denominator:
-3 * x > -3 * [(15 - 2x) / -3]
-3x > 15 - 2x
Next, we can bring all terms involving x to one side:
-3x + 2x > 15
-x > 15
Now, if we multiply both sides by -1, we reverse the inequality sign:
x < -15
So, the solution to the inequality x > (15 - 2x) / -3 is x < -15.
To write this in set builder notation, we can express the solution as { x | x < -15 }. This reads as "the set of all x such that x is less than -15".
2. To solve the system of equations:
y + x = 9 ...(equation 1)
3y = 2x + 8 ...(equation 2)
There are several methods to solve a system of equations, but one common method is substitution.
First, solve equation 1 for y in terms of x:
y = 9 - x
Now, substitute this value of y into equation 2:
3(9 - x) = 2x + 8
Simplify and solve for x:
27 - 3x = 2x + 8
Combine like terms:
27 = 5x + 8
Subtract 8 from both sides:
19 = 5x
Divide by 5:
x = 19/5
Now, substitute this value of x back into equation 1 to find y:
y + (19/5) = 9
Multiply through by 5 to eliminate the fraction:
5y + 19 = 45
Subtract 19 from both sides:
5y = 45 - 19
5y = 26
Divide by 5:
y = 26/5
Therefore, the solution to the system of equations y + x = 9 and 3y = 2x + 8 is x = 19/5 and y = 26/5.
3. To find the maximum y-value of the quadratic function f(x) = -2x^2 + 7x + 1, we can identify the vertex of the parabola.
The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).
In this case, a = -2 and b = 7.
x = -7 / (2 * -2)
x = -7 / -4
x = 7/4
To find the maximum y-value, substitute this x-value back into the original function:
f(7/4) = -2(7/4)^2 + 7(7/4) + 1
Simplify this expression to find the maximum y-value.
4. To solve the equation 4x/3 + 1/2 = 5y/6, we can follow these steps:
Step 1: Multiply through by the least common denominator (LCD) to eliminate fractions. The LCD in this case is 6.
6*(4x/3) + 6*(1/2) = 6*(5y/6)
8x + 3 = 5y
Step 2: Solve for y in terms of x:
5y = 8x + 3
Divide both sides by 5:
y = (8x + 3)/5
So the solution to the equation is y = (8x + 3)/5.
5. To find the solution set to the equation 2 |x - 100| = 50 + |x - 100|, we can split it into two cases based on the sign of (x - 100).
Case 1: (x - 100) ≥ 0
In this case, we can remove the absolute value brackets without changing the equality:
2(x - 100) = 50 + (x - 100)
Simplify and solve for x:
2x - 200 = 50 + x - 100
x = 150
Case 2: (x - 100) < 0
In this case, we need to change the sign inside the absolute value brackets:
2(-x + 100) = 50 + (-x + 100)
Simplify and solve for x:
-2x + 200 = 50 - x + 100
Combine like terms:
-x + 250 = -x + 100
Since the -x terms cancel each other out, we are left with:
250 = 100
However, this is not a true statement. So, there are no solutions in this case.
Therefore, the solution set to the equation is x = 150.