Deck 17: Nonlinear Inequalities

Question 1

Determine whether the value of $x = - 6$ is a solution of the inequality.​ $x ^ { 2 } - x - 14 \geq 0$ ​

A)No
B)Yes

Answer 1

Yes

Question 2

Find the key numbers of the expression.​ $\frac { 1 } { x - 7 } + 1$ ​

A)-6,7
B)6,7,1
C)-6,-7
D)6,7
E)6,-7

Answer 2

6,7

Question 3

Solve the inequality and graph the solution on the real number line.​ $$x ^ { 2 } > 2 x + 48$$ ​

A) $$( - \infty , - 6 ) \cup ( 8 , \infty )$$
B) $$( - \infty , 6 ) \cup ( 8 , \infty )$$
C) $$( - \infty , - 6 ) \cup ( - 8 , - \infty )$$
D) $$( - \infty , - 6 ) \cup ( - 8 , \infty )$$
E) $$( - \infty , 6 ) \cup ( 8 , \infty )$$

Answer 3

$$( - \infty , - 6 ) \cup ( 8 , \infty )$$

Question 4

Solve the inequality and write the solution set in interval notation.​ $x ^ { 3 } - 36 x \geq 0$ ​

A) $[ - 6,0 ] \cup [ 6 , \infty ]$
B) $[ - 6,0 ] \cup [ 6 , \infty )$
C) $[ - 36,0 ] \cap [ 6 , \infty ]$
D) $[ - 36,0 ] \cup [ - 6 , \infty )$
E) $[ - 6,0 ] \cap [ 6 , \infty )$

Question 5

Determine whether the value of $x = - 1$ is a solution of the inequality.​ $\frac { 3 x ^ { 2 } } { x ^ { 2 } + 4 } < 1$ ​

A)Yes
B)No

Question 6

Solve the inequality and write the solution set in interval notation.​ $6 x ^ { 2 } - 10 x < 0$ ​

A) $$( - \infty , 0 ) ] \cup [$$ $\frac { 5 } { 3 }$ $, \infty )$
B) $$\varnothing$$
C) $$( - \infty , 0 ) \cup($$ $\frac { 5 } { 3 }$ $, \infty )$
D) $( \infty , 0 ) \cup [ -$ $\frac { 5 } { 3 }$ $, \infty )$
E) $( 0 ,$ $\frac { 5 } { 3 }$ ) ​

Question 7

Solve the inequality and graph the solution on the real number line​ $$x ^ { 3 } + 3 x ^ { 2 } - 9 x - 27 \leq 0$$ ​

A) $$( - \infty , 3 ]$$
B) $$( - \infty , 3 )$$
C) $$( - \infty , 3 ]$$
D) $$( - \infty , - 3 ]$$
E) $$( - \infty , - 3 )$$

Question 8

Determine whether the value of $$x = \frac { 2 } { 7 }$$ is a solution of the inequality. $x ^ { 2 } - 2 < 0$ ​

A)Yes
B)No

Question 9

Solve the inequality and graph the solution on the real number line​ $x ^ { 2 } \leq 9$ ​

A) $( - 3,3 )$
B)​ $[ - 3,3 ]$
C) $[ - 3,3 ]$
D) $[ 3 , - 3 ]$
E) $( - 3,3 )$

Question 10

Solve the inequality and graph the solution on the real number line.​ $$x ^ { 2 } + 5 x + 13 > 0$$ ​

A) $$x \geq - 13$$
B) $$x \leq 13$$
C) $$x > 5$$
D) $$x < - 5$$
E)All real numbers

Question 11

Solve the inequality and graph the solution on the real number line.​ $9 x ^ { 2 } - 6 x + 1 \leq 0$ ​

A)​ $x = 3$
B) $x = \frac { 1 } { 3 }$
C) $x = \frac { 1 } { 9 }$
D) $x = - \frac { 1 } { 3 }$
E) $x = - 3$

Question 12

Solve the inequality and graph the solution on the real number line.​ $$x ^ { 2 } - 10 x - 24 > 0$$ ​

A) $$( - \infty , - 2 ) \cup ( 12 , \infty )$$
B) $$( - \infty , - 2 ) \cap ( - 12 , \infty )$$
C) $$( - \infty , - 2 ) \cup ( - \infty , - 12 , )$$
D) $$( \infty , - 2 ) \cup [ 12 , \infty )$$
E) $$( - \infty , 2 ] \cup ( 12 , \infty )$$

Question 13

Solve the inequality and graph the solution on the real number line.​ $$x ^ { 3 } - 3 x ^ { 2 } - x > - 3$$ ​

A) $$( - 1,1 ) \cup ( 3 , \infty )$$
B) $$( - 1,1 ) \cup ( - 3 , \infty )$$
C) $$( - 1,1 ) \cup ( 3 , \infty )$$
D) $$( - 1,1 ) \cup ( - 3 , - \infty )$$
E) $$( - 1,1 ) \cup ( - 3 , \infty )$$

Question 14

Solve the inequality and graph the solution on the real number line.​ $$x ^ { 2 } + 3 x > 4$$ ​

A) $$( - \infty , - 4 ) \cup ( 1 , \infty )$$
B) $$( - \infty , - 4 ) \cup ( - 1 , \infty )$$
C) $$( - \infty , - 4 ) \cup ( 1 , \infty )$$
D) $$( - \infty , - 4 ) \cup ( - 1 , \infty )$$
E) $$( - \infty , - 4 ) \cup ( 1 , - \infty )$$

Question 15

Find the key numbers of the expression.​ $16 x ^ { 3 } - 49 x ^ { 2 }$ ​

A)0, $\frac { 16 } { 49 }$
B)0,49
C)0,51
D)0,-49
E)0, $\frac { 49 } { 16 }$

Question 16

Solve the inequality and graph the solution on the real number line. ​​ $( x - 7 ) ^ { 2 } \geq 1$ ​

A) $( - \infty , - 6 ] \cup [ - 8 , \infty )$
B) $( - \infty , 6 ] \cup [ 8 , \infty )$
C) $( - \infty , 8 ] \cup [ 6 , \infty )$
D) $( - \infty , 8 ] \cup [ 6 , \infty )$
E) $( - \infty , 6 ] \cup [ 8 , \infty )$

Question 17

Solve the inequality and graph the solution on the real number line.​ $x ^ { 2 } - 10 x + 8 < 19$ ​

A) $( - 1,11 )$
B) $( - 1 , - 11 )$ ​
C) $( 1 , - 11 )$ ​
D) $( - 1 , - 11 )$
E) $( - 1,11 )$

Question 18

Solve the inequality and graph the solution on the real number line.​ $x ^ { 2 } + x < 12$ ​

A) $( - 4,3 )$
B) $( - 4,3 )$
C) $( 4,3 )$
D) $( - 4 , - 3 )$
E) $( - 4 , - 3 )$

Question 19

Solve the inequality and write the solution set in interval notation.​ $5 x ^ { 3 } - 20 x ^ { 2 } > 0$ ​

A) $( 4 , \infty )$
B) $( - 4 , - \infty )$
C) $( - 4 , \infty )$
D) $( 4 , - \infty )$
E) $( - \infty , 5 )$

Question 20

Solve the inequality and graph the solution on the real number line.​ $x ^ { 2 } < 4$ ​

A) $[ - 2,2 ]$
B) $[ - 2,2 ]$
C) $( - 2,2 )$ ​
D) $( - 2,2 )$ ​
E) $( - 2,2 )$ ​

Question 21

Solve the inequality and write the solution set in interval notation.​ $x ^ { 8 } ( x - 2 ) \leq 0$ ​ ​

A) $( - \infty , 2 ]$
B) $( - \infty , - 2 ) ]$
C) $( - \infty , 2 )$
D) $( - 2,2 ]$

Question 22

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy each inequality. ​
Equation: $y = \frac { 6 x } { x - 5 }$ ​
Inequality: $y \geq 8$ ​

A)​ $$5 < x \leq - \frac { 10 } { 1 }$$
B)​ $$5 < x \leq \frac { 10 } { 1 }$$
C)​ $$5 < x$$
D)​ $$5 < x \leq \frac { 10 } { 1 }$$
E)​ $$6 < x \leq \frac { 10 } { 1 }$$

Question 23

Find the domain of x in the expression.​ $\sqrt { 25 - x ^ { 2 } }$ ​

A) $( - 25,25 )$
B) $$[ - 25,25 ]$$
C) $$( - 5,5 )$$
D) $[ - 5,25 )$
E) $[ - 5,5 ]$

Question 24

Solve the inequality and graph the solution on the real number line.​ $$\frac { 3 x - 5 } { x - 5 } \geq 0$$ ​

A)( $$- \infty , \frac { 5 } { 3 }$$ ​] $$\cup( 5 , \infty )$$
B)( $$- \infty , \frac { 5 } { 3 }$$ ​] $$\cup ( - 5 , \infty )$$
C)( $$- \infty , - \frac { 5 } { 3 }$$ ​] $$\cup( 5 , \infty )$$
D)( $$- \infty , - \frac { 5 } { 3 }$$ ​] $$\cup ( - 5 , \infty )$$
E)( $$- \infty , \frac { 5 } { 3 }$$ ​] $$\cap ( 5 , \infty )$$

Question 25

Solve the inequality and graph the solution on the real number line.​ $$\frac { 2 } { x + 4 } > \frac { 1 } { x - 2 }$$ ​

A) $$( - 4,2 ) \cup ( 8 , \infty )$$
B) $$( - 4,2 ) \cap ( 8 , \infty )$$
C) $$( - 4 , - 2 ) \cup ( 8 , \infty )$$
D) $$( - 4,2 ) \cap ( - 8 , \infty )$$
E) $$( 4,2 ) \cup ( 8 , \infty )$$

Question 26

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy the following inequality. ​
Equation: $y = \frac { 7 x } { x ^ { 2 } + 6 }$ ​
Inequality: $y \geq 1$ ​

A)​ $$1 \leq x \leq 6$$
B)​ $$1 \leq x < 6$$
C)​ $$1 < x < 6$$
D)​ $$1 < x \leq 6$$
E)​ $$1 < x \leq 6$$

Question 27

Solve the inequality and graph the solution on the real number line.​ $$\frac { 4 x - 1 } { x } > 0$$ ​

A) $$( - \infty , 0 ) \cup \left( \frac { 1 } { 4 } , \infty \right)$$
B) $$( - \infty , 0 ) \cup \left( - \frac { 1 } { 4 } , \infty \right)$$ ​
C) $$( - \infty , 0 ) \cap \left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)$$
D) $$( - \infty , 0 ) \cap \left( \frac { 1 } { 4 } , \infty \right)$$
E) $$( - \infty , 0 ) \cap \left( - \frac { 1 } { 4 } , \infty \right)$$

Question 28

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy each inequality. ​
Equation: $y = \frac { 6 ( x - 6 ) } { x + 5 }$ ​
Inequality: $y \leq 0$ ​

A)​ ​ $$- 5 \leq x \leq 6$$
B)​ $$- 5 < x > 6$$
C)​ $$- 5 < x \leq 6$$ ​
D) $$- 5 < x \leq 6$$
E)​ $$- 5 < x < 6$$

Question 29

Find the domain of x in the expression.​ $\sqrt { x ^ { 2 } - 7 x + 12 }$ ​

A) $( - \infty , 3 ] \cup [ 4 , \infty )$
B) $( - \infty , 3 ] \cap [ 4 , \infty )$
C) $( \infty , - 3 ] \cup [ 4 , \infty )$
D) $( \infty , 3 ] \cap [ 4 , \infty )$
E) $( \infty , - 3 ] \cap [ 4 , \infty )$

Question 30

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy each inequality. ​
Equation: $y = x ^ { 3 } - x ^ { 2 } - 16 x + 16$ ​
Inequality: $y \leq 0$ ​

A)​ ​ $$\begin{array} { l } \infty < x \leq 4 \\1 \leq x \leq 4\end{array}$$
B)​ ​ $$\begin{array} { l } \infty < x \leq - 4 \\1 \leq x \leq 4\end{array}$$
C)​ ​ $$\begin{array} { l } - \infty < x \leq - 4 \\- 1 \leq x \leq - 4\end{array}$$
D)​ $$\begin{array} { l } - \infty < x \leq - 4 \\1 \leq x \leq 4\end{array}$$
E)​ ​ $$- \infty < x \leq 4$$

Question 31

Solve the inequality and graph the solution on the real number line.​ $$\frac { x + 9 } { x + 1 } - 2 < 0$$ ​

A) $$( - \infty , - 1 ) \cup ( 7 , \infty )$$
B) $$( - \infty , - 1 ) \cap ( - 7 , \infty )$$
C) $$( - \infty , 1 ) \cap ( 7 , \infty )$$
D) $$( - \infty , - 1 ) \cup ( - 7 , \infty )$$
E) $$( - \infty , - 1 ) \cap ( 7 , \infty )$$

Question 32

Find the domain of x in the expression.​ $\sqrt { \frac { x } { x ^ { 2 } - 36 } }$ ​

A) $( - 36,0 ] \cup ( 36 , \infty )$
B) $( - 36,0 ] \cap ( 36 , \infty )$
C) $( - 36,0 ] \cap ( 6 , \infty )$
D) $( - 6,0 ] \cup ( 6 , \infty )$
E) $( - 36,0 ] \cap ( - 6 , \infty )$

Question 33

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy each inequality. ​
Equation: $$y = \frac { 5 x ^ { 2 } } { x ^ { 2 } + 25 }$$ ​
Inequality: $$y \leq 2$$ ​

A)​ $$- \infty < x \leq \frac { 5 \sqrt { 3 } } { 3 }$$
B)​ $$- \infty < x < \infty$$
C)​ $$- \frac { 5 \sqrt { 3 } } { 3 } \leq x \leq \frac { 5 \sqrt { 3 } } { 3 }$$
D)​ $$- \infty < x \leq \frac { \sqrt { 3 } } { 3 }$$
E)​ $$- \infty < x \leq 5 \sqrt { 3 }$$

Question 34

Solve the inequality and graph the solution on the real number line.​ $$\frac { x ^ { 2 } + 4 x } { x ^ { 2 } - 25 } \leq 0$$ ​

A) $$( - 5 , - 4 ] \cup [ 0,5 )$$
B) $$( - 5,4 ] \cup [ 0 , - 5 )$$
C) $$( - 5,4 ] \cup [ 0,5 )$$
D) $$( 5 , - 4 ] \cup [ 0,5 )$$
E) $$( 5,4 ] \cup [ 0,5 )$$

Question 35

Solve the inequality and graph the solution on the real number line.​ $$\frac { 1 } { x } \geq \frac { 1 } { x + 2 }$$ ​

A) $$( - \infty , - 2 ) \cap ( 0 , \infty )$$ ​
B) $$( - \infty , - 2 ) \cup ( 0 , \infty )$$
C) $$( - \infty , 2 ) \cap ( 0 , \infty )$$
D) $$( \infty , 2 ) \cup ( 0 , \infty )$$
E) $$( - \infty , 2 ) \cup ( 0 , \infty )$$

Question 36

Find the domain of x in the expression.​ $\sqrt { \frac { x } { x ^ { 2 } - 2 x - 24 } }$ ​

A) $( - 4,0 ] \cap ( 6 , \infty )$
B) $( - 6,0 ] \cup ( 4 , \infty )$
C) $( - 4,0 ] \cup ( 6 , \infty )$
D) $( - 4,0 ] \cup ( - 6 , \infty )$
E) $( - 6,0 ] \cap ( - 4 , \infty )$

Question 37

Find the domain of x in the expression.​ $\sqrt { 9 - 4 x ^ { 2 } }$ ​

A) $\left[ \frac { 3 } { 2 } , \frac { 3 } { 2 } \right]$
B) $\left[ - \frac { 3 } { 2 } , - \frac { 3 } { 2 } \right]$
C) $\left[ - \frac { 3 } { 2 } , \frac { 3 } { 2 } \right]$
D) $\left[ - \frac { 2 } { 3 } , \frac { 2 } { 3 } \right]$

Question 38

Solve the inequality and graph the solution on the real number line.​ $$\frac { x ^ { 2 } + x - 12 } { x } \geq 0$$ ​

A) $$[ - 4,0 ) \cap [ 3 , \infty )$$
B) $$[ 4,0 ) \cup [ 3 , \infty )$$
C) $$[ 4,0 ) \cup [ - 3 , \infty )$$
D) $$[ - 4,0 ) \cap [ - 3 , \infty )$$
E) $$[ - 4,0 ) \cup [ 3 , \infty )$$

Question 39

Use a graphing utility to graph the equation.Use the graph to approximate the values of x that satisfy the inequality. ​
Equation: $y = \frac { 1 } { 6 } x ^ { 2 } - 6 x + 1$ ​
Inequality: $y \leq 0$ ​

A)​ $$x \leq - 1 - \sqrt { \frac { 4 } { 27 } } , 1 - \sqrt { \frac { 4 } { 27 } } \leq x \leq 1 + \sqrt { \frac { 4 } { 27 } }$$
B)​ $$1 - \sqrt { \frac { 4 } { 27 } } \leq \mathrm { x } \leq 1 + \sqrt { \frac { 4 } { 27 } }$$
C)​ $$x \leq 1 - \sqrt { \frac { 4 } { 27 } } , 1 + \sqrt { \frac { 4 } { 27 } } \leq x$$
D)​ $$x \leq - 1 - \sqrt { \frac { 4 } { 27 } } , - 1 + \sqrt { \frac { 4 } { 27 } } \leq x$$
E)​ $$- 1 - \sqrt { \frac { 4 } { 27 } } \leq x \leq - 1 + \sqrt { \frac { 4 } { 27 } }$$

Question 40

Find the domain of x in the expression.​ $\sqrt { x ^ { 2 } - 4 }$ ​

A) $( - \infty , 2 ] \cap [ 2 , \infty )$
B) $( - \infty , - 2 ] \cup [ 2 , \infty )$
C) $( - \infty , - 2 ] \cap [ 2 , \infty )$
D) $( - \infty , 2 ] \cup [ 2 , \infty )$
E) $( - \infty , - 2 ) \cup [ 2 , \infty )$

Question 41

The numbers N (in millions)of students enrolled in schools in the United States from 1995 through 1997 are shown in the table. ​
$$\begin{array} { | c | c | } \hline \text { Year } & \text { Number, } N \\\hline 1995 & 71.2 \\\hline 1996 & 72.3 \\\hline 1997 & 74 \\\hline\end{array}$$
Use a graphing utility to create a scatter plot of the data.Let t represent the year,with t = 5 corresponding to 1995.

A)​ ​
B)​
C)​
D)​
E)​

Question 42

Determine the intervals on which the following polynomial is entirely negative and those on which it is entirely positive. $- x ^ { 2 } - 2 x + 35$

A)entirely negative: $( - \infty , 6 )$ ;entirely positive: $( 6 , \infty )$
B)entirely negative: $( - \infty , - 7 ) , ( 5,6 )$ ;entirely positive: $( - 7,6 )$
C)entirely negative: $( - \infty , - 7 ) , ( 5 , \infty )$ ;entirely positive: $( - 7,5 )$
D)entirely negative: $( - 7,6 )$ ;entirely positive: $( 0 , \infty )$
E)entirely negative: $( - \infty , 0 )$ ;entirely positive: $( 0 , \infty )$

Question 43

Find the domain of x in the expression $\sqrt { 16 - x ^ { 2 } }$ .

A) $( - \infty , - 4 ] \cup [ 4 , \infty )$
B) $( - \infty , 4 ]$
C) $[ - 4,4 ]$
D) $[ - 4 , \infty )$
E) $( - \infty , \infty )$

Question 44

Find the domain of x in the expression $\sqrt [ 4 ] { x ^ { 3 } - 3 x }$ . ​

A) $( - \infty , - \sqrt { 3 } ] , [ \sqrt { 3 } , \infty )$
B) $( - \infty , \sqrt { 3 } ] , [ \sqrt { 3 } , \infty )$
C) $[ - \sqrt { 3 } , 0 ] , [ \sqrt { 3 } , \infty )$
D) $[ - \sqrt { 3 } , \sqrt { 3 } ]$
E) $( - \infty , - \sqrt { 3 } ) , ( 0 , \sqrt { 3 } )$

Question 45

Determine whether the statement is true or false.Justify your answer, the solution set of the inequality $\frac { 7 } { 2 } x ^ { 2 } + 7 x + 14 \geq 0$ is the entire set of real numbers.
​

A)False,the y -values are less than zero for all values of x.
B)True,the y -values are greater than zero for all values of x.

Question 46

Use the position equation $s = - 16 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ ,where s represents the height of an object (in feet), $v _ { 0 }$ represents the initial velocity of the object (in feet per second), $s_0$ represents the initial height of the object (in feet),and t represents the time (in seconds). A projectile is fired straight upward from ground level $\left( s _ { 0 } = 0 \right)$ with an initial velocity of 160 feet per second.At what instant will it be back at ground level?
​

A) $t = 8$ Sec
B) $t = 10$ Sec
C) $t = 12$ Sec
D) $t = 11$ Sec
E) $t = 9$ Sec

Question 47

Find the interval(s)for b such that the following equation has at least one real solution​ $x ^ { 2 } + b x + 6 = 0$ . ​

A) $( - \infty , - \sqrt { 24 } ] \cup [ \sqrt { 24 } , \infty )$
B) $( - \infty , - \sqrt { 24 } ] \cap [ \sqrt { 24 } , \infty )$
C) $( - \infty , \sqrt { 24 } ] \cup ( \sqrt { 24 } , \infty )$
D) $( - \infty , - \sqrt { 24 } ) \cup [ - \sqrt { 24 } , \infty )$
E) $( - \infty , \sqrt { 24 } ] \cap [ \sqrt { 24 } , \infty )$

Question 48

Solve the inequality $36 x - x ^ { 3 } < 0$ and write the solution set in interval notation.

A) $( - \infty , - 6 ) \cup ( 0,6 )$
B) $( - 6,6 )$
C) $( - \infty , 6 )$
D) $( - 6,0 ) \cup ( 6 , \infty )$
E) $( - \infty , \infty )$

Question 49

A rectangular parking lot with a perimeter of 442 feet is to have an area of at least 8050 square feet.Within what bounds must the length of the rectangle lie? ​

A) $8050 \mathrm { ft } \leq L \leq 442 \mathrm { ft }$
B) $$177 \mathrm { ft } \leq L \leq 48 \mathrm { ft }$$
C) $$49 \mathrm { ft } \leq L \leq 178 \mathrm { ft }$$
D) $8050 \mathrm { ft } \leq L \leq 47 \mathrm { ft }$
E) $46 \mathrm { ft } \leq L \leq 175 \mathrm { ft }$

Question 50

When two resistors of resistances $R _ { 1 }$ and $R _ { 2 }$ are connected in parallel (see figure),the total resistance $R$ satisfies the equation​ $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } }$ . Find $R _ { 1 }$ for a parallel circuit in which $R _ { 2 } = 7$ ohms and R must be at least 1 ohm. ​

A) $R _ { 1 } \leq 7 \text { ohms }$
B) $R _ { 1 } < 7 \text { ohms }$
C) $R _ { 1 } \geq$ $\frac { 7 } { 6 }$ $\text { ohms }$
D) $R _ { 1 } > 7 \mathrm { ohms }$
E) $R _ { 1 } =$ $\frac { 7 } { 6 }$ $\text { ohms }$

Question 51

A rectangular playing field with a perimeter of 98 meters is to have an area of at least 444 square meters.Within what bounds must the length of the rectangle lie?
​

A) $444 m \leq L \leq 13 m$
B) $37 m \leq L \leq 98 m$
C) $12 m \leq L \leq 37 m$
D) $39 m \leq L \leq 14 m$
E) $444 m < L < 13 m$

Question 52

Solve: $x ^ { 2 } - 2 x - 15 < 0$

A) $( - \infty , - 2 )$
B) $( - 3 , \infty )$
C) $( - 3,5 )$
D) $( - \infty , 5 )$
E) $( 5 , \infty )$

Question 53

Find the domain of x in the following expression. $\sqrt [ 3 ] { x ^ { 4 } - 625 }$

A) $[ - 5,5 ]$
B) $( - \infty , 5 ) \cup ( 5 , \infty )$
C) $[ 5 , \infty )$
D) $( 5 , \infty )$
E)all real numbers

Question 54

Find the interval(s)for b such that the following equation has at least one real solution and write a conjecture about the interval(s)based on the values of the coefficients​ $4 x ^ { 2 } + b x + 13 = 0$ . ​

A) $( - \infty , - 2 \sqrt { 52 } ] \cup [ 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$
B) $( - \infty , - 2 \sqrt { 52 } ] \cup [ - 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c < 0$ , $b \leq - 2 \sqrt { a c }$ or $$b \geq 2 \sqrt { a c }$$
C) $( - \infty , 2 \sqrt { 52 } ] \cup [ 2 \sqrt { 52 } , \infty )$ If $a > 0$ and $c > 0$ , $- b \leq - 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$
D) $( - \infty , 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $$a < 0$$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$
E) $( - \infty , - 2 \sqrt { 52 } ] \cap [ 2 \sqrt { 52 } , \infty )$ If $a < 0$ and $c < 0$ , $b \leq 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$

Question 55

Find the interval(s)for b such that the equation has at least one real solution and write a conjecture about the interval(s)based on the values of the coefficients​ $4 x ^ { 2 } + b x + 7 = 0$ . ​

A) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c < 0$ , $b \geq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$
B) $( - \infty , - 2 \sqrt { 28 } ] \cup [ - 2 \sqrt { 28 } , \infty )$ If $a < 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$
C) $( - \infty , - 2 \sqrt { 28 } ] \cup [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq - 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$
D) $( - \infty , 2 \sqrt { 28 } ] \cup [ 2 \sqrt { 28 } , \infty )$ If $a > 0$ and $c > 0$ , $b \leq 2 \sqrt { a c }$ or $b \geq 2 \sqrt { a c }$
E) $( - \infty , - 2 \sqrt { 28 } ] \cap [ 2 \sqrt { 28 } , \infty )$ If $a < 0$ and $c < 0$ , $b \leq 2 \sqrt { a c }$ or $b \leq 2 \sqrt { a c }$

Question 56

Use the position equation $s = - 16 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ ,where $S$ represents the height of an object (in feet), $v_0$ represents the initial velocity of the object (in feet per second), $s_0$ represents the initial height of the object (in feet),and t represents the time (in seconds). A projectile is fired straight upward from ground level $\left( s _ { 0 } = 0 \right)$ with an initial velocity of 176 feet per second.At what instant will it be back at ground level?
​

A) $t = 11$ Sec
B) $t = 16$ Sec
C) $t = 13$ Sec
D) $t = 12$ Sec
E) $t = 10$ Sec