Help with these questions
help with these questions~
Thank you
1. Determine the interval(s) on which x^2 + 2x -3>0
a)x<-3, x>1
b)-3<x<1
c)x<-3, -3<x<1, x>1
d) x>1
Answer is D, x>1
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2. Determine when the function f(x)= 3x^3 + 4x^2 -59x -13 is greater than 7.
a) -5<x<1/3
b) -5<x<-1/3, x>4
c) x<4
d) x=-5, -1/3,4
Answer is: B
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3. Provde the intervals you would check to determine when -5x^2 + 37x> -15x^2 + 12x + 15.
a) x=3, x=-0.5
b) x<-3, -3<x<0.5, x>0.5
c) x<-1, -1<x<15, x>15
d) x= 1, x=15
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1. x^2 + 2x - 3 > 0
(x+3)(x-1) > 0
x-intercepts are -3 and 1
so you are looking for the values of x when the parabola y = x^2 + 2x - 3 is above the x-axis
Since it opens upwards those values are
x < -3 OR x > 1
The closest choice to that is #1, but they did not include the necessary OR. The comma is this context means AND, which would be incorrect.
2. 3x^3 + 4x^2 - 59x - 13 > 7
3x^3 + 4x^2 - 59x - 20 > 0
after some quick tries of ±1, ±2, ± 4, I found x=4 to be a solution, so x-4 is a factor
by synthetic division,
3x^3 + 4x^2 - 59x - 20 = (x-4)(3x^2 + 16x + 5)
= (x-4)(x+5)(3x+1)
so the critical values are -5, -1/3, and 4
So the curve is above the x-axis (or above 7 in the original) for
-5 < x < -1/3 OR x > 4
Again, they made an error by not stating the OR condition, but it looks like they meant b) , which is what you had.
3. -5x^2 + 37x> -15x^2 + 12x + 15
10x^2 + 25x - 15 > 0
2x^2 + 5x - 3 > 0
(2x - 1)(x + 3) > 0
critical values are 1/s and -3
so I would check
x < -3, -3 < x < 1/2, and x > 1/2
looks like b) is the correct choice.
1. The graph is a parabola, opening upward. So, there will be a left side and a right side above the x-axis. These intervals will be outside the two roots.
A is the correct answer
2. B is correct
3. The graphs intersect at x = -3 and 0.5
So, B is the answer
Thank you~
1. To determine the interval(s) on which x^2 + 2x - 3 > 0, we need to find the values of x that make the expression greater than 0.
First, let's factor the expression: x^2 + 2x - 3 = (x + 3)(x - 1)
Now, we have two factors (x + 3) and (x - 1) which represent the points where the expression becomes zero. When x is less than -3, the expression (x + 3) is negative, and when x is greater than 1, the expression (x - 1) is positive.
Therefore, the answer is x > 1. Option D is the correct answer.
2. To determine when the function f(x) = 3x^3 + 4x^2 - 59x - 13 is greater than 7, we need to find the values of x that make the function greater than 7.
First, let's set up the inequality: 3x^3 + 4x^2 - 59x - 13 > 7
Now, we need to solve this inequality. To do that, we can subtract 7 from both sides: 3x^3 + 4x^2 - 59x - 20 > 0
Next, let's factor the expression: (x + 4)(3x - 1)(x - 5) > 0
Now, we have three factors (x + 4), (3x - 1), and (x - 5) which represent the points where the expression becomes zero. When x is less than -4 or between 1/3 and 5, the expression is positive.
Therefore, the answer is -5 < x < 1/3 or x > 4. Option B is the correct answer.
3. To determine the intervals where -5x^2 + 37x > -15x^2 + 12x + 15, we need to find the values of x that make the expression greater than zero.
First, let's simplify the equation: 10x^2 + 25x - 15 > 0
Now, we need to solve this inequality. To do that, we can divide both sides by 5 to get: 2x^2 + 5x - 3 > 0
Next, let's factor the expression: (2x - 1)(x + 3) > 0
Now, we have two factors (2x - 1) and (x + 3) which represent the points where the expression becomes zero. When x is less than 1/2 or greater than -3, the expression is positive.
Therefore, the answer is x < 1/2 or x > -3. Option B is the correct answer.