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. Correct.

2. Correct.

3. -5x^2 + 37x > -5x^2 + 12x + 15.
-5x^2 + 5x^2 + 37x - 12x > 15,
Combine like-terms:
25x > 15,
X > 15/25,
X > 3/5.

Ans. = D.

Correction: The -5x^2 on the right side of the inequality sign should be

-15x^2.

To determine the intervals on which an inequality is true, such as x^2 + 2x - 3 > 0, you can use the following steps:

1. Factorize the quadratic expression: x^2 + 2x - 3 = (x + 3)(x - 1)

2. Set each factor equal to zero to find the critical values:
x + 3 = 0 => x = -3
x - 1 = 0 => x = 1

3. Plot these critical values on a number line, dividing it into intervals:
-∞ -3 1 ∞

4. Choose a value within each interval and substitute it back into the original inequality.
For example, let's choose x = 0:
(0)^2 + 2(0) - 3 = -3

5. Determine whether the inequality is true or false for each interval based on the chosen values. In this case, the inequality is true only when x > 1, as indicated by the answer choice "D".

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To determine when the function f(x) = 3x^3 + 4x^2 - 59x - 13 is greater than 7, you can follow these steps:

1. Set the function equal to 7: 3x^3 + 4x^2 - 59x - 13 > 7

2. Re-arrange the equation to form a quadratic expression: 3x^3 + 4x^2 - 59x - 20 > 0

3. You can use a similar approach as in the previous question to find the intervals where the inequality is true.

4. Factorize the quadratic expression: 3x^3 + 4x^2 - 59x - 20 = (x + 5)(x + 1/3)(3x - 4)

5. Set each factor equal to zero to find the critical values:
x + 5 = 0 => x = -5
x + 1/3 = 0 => x = -1/3
3x - 4 = 0 => x = 4/3

6. Plot the critical values on a number line, dividing it into intervals:
-∞ -5 -1/3 4/3 ∞

7. Choose a value within each interval and substitute it back into the original inequality. For example, let's choose x = 0:
3(0)^3 + 4(0)^2 - 59(0) - 20 = -20

8. Determine whether the inequality is true or false for each interval based on the chosen values. In this case, the inequality is true for two intervals: -5 < x < -1/3 and x > 4/3, as indicated by the answer choice "B".

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To determine the intervals to check when -5x^2 + 37x > -15x^2 + 12x + 15, you can follow these steps:

1. Subtract the expressions on both sides of the inequality:
-5x^2 + 37x - (-15x^2 + 12x + 15) > 0
Simplify it: -5x^2 + 37x + 15x^2 - 12x - 15 > 0
Combine like terms: 10x^2 + 25x - 15 > 0

2. Factorize the quadratic expression: 10x^2 + 25x - 15 = 5(2x^2 + 5x - 3)

3. Set each factor equal to zero to find the critical values:
2x^2 + 5x - 3 = 0

4. Use the quadratic formula or factoring techniques to solve the equation:
(2x - 1)(x + 3) = 0
x = 1/2, -3

5. Plot the critical values on a number line, dividing it into intervals:
-∞ -3 1/2 ∞

6. Substitute a value from each interval back into the original inequality to determine whether it is true or false. For example, let's choose x = 0:
-5(0)^2 + 37(0) - (-15(0)^2 + 12(0) + 15) = 0 + 0 - (0 + 0 + 15) = -15

7. Determine the intervals where the inequality is true or false based on the chosen values. In this case, the inequality is true for -∞ < x < -3 and 1/2 < x < ∞, as indicated by the answer choice "B".