Subtract (enter answer is standard form)
(-2x^4-2x^3+8x^2+2)-(x^4-5x^3+2x+7)

Which graph represents the inequality? (has picture)
2x-3y≤6

Multiply
8y^4(2y^4-3y^3+5y)

Solve for x
0=7x^2+x+5

Harold and Helen both spend 3 hours reading a novel. Helen reads 30 more pages than Harold. Helen reads at least 40 pages in 1 hour, but no more than 70 pages in 1 hour.
Which statement correctly describes the range of pages Harold reads?


Harold reads more than 90 pages, but less than 180 pages.

Harold reads at least 90 pages, but no more than 180 pages.

Harold reads more than 150 pages, but less than 240 pages.

Harold reads at least 150 pages, but no more than 240 pages.

PB&P sells peanut butter and pickle sandwiches. The Standard special sells for $2 and the Deluxe special sells for $4.50. When all related business expenses are included, the Standard special costs $0.50 to prepare and the Deluxe special costs $1.25 to prepare.
Last Monday, PB&P sold at least $200 worth of Standard and Deluxe peanut butter and pickle sandwich specials and its expenses were no more than $100. At least 30 Standard special were sold.
Let x be the number of Standard Specials sold last Monday and y be the number of Deluxe specials sold last Monday.
Which ordered pairs representing a combination of Standard specials and Deluxe specials could have been sold last Monday and make sense in the context of the situation?
Select each correct answer.

(30,68)(30,68)

(80,40.5)(80,40.5)

(60,70)(60,70)

(40,60)(40,60)

(50.5,40)

Effie is selling candles to raise money for new soccer league equipment. A small candle sells for $8 and a large candle sells for $12. The number of large candles Effie sells must be greater than or equal to 2 times the number of small candles she sells. She has at most 48 candles to sell.
The number of small candles sold is represented by x and the number of large candles sold is represented by y.
What is the maximum revenue she can make?

$448

$512

$576

$588

What is the product of 4−5i and 2+3i?
Enter your answer, in standard form, in the box.

Answer:

Part 1) [tex]-3x^4+3x^3+8x^2-2x-5[/tex]

Part 2) The graph in the attached  figure

Part 3) [tex]16y^8-24y^7+40y^5[/tex]

Part 4) [tex]x1=\frac{-1+\sqrt{139}i} {14}[/tex]  and [tex]x2=\frac{-1-\sqrt{139}i} {14}[/tex]

Part 5) Harold reads more than [tex]90[/tex] pages, but less than [tex]180[/tex] page

Part 6)  [tex](30,68)[/tex], [tex](40,60)[/tex]

Part 7) [tex]\$576[/tex]

Part 8) [tex]23 + 2i[/tex]

Step-by-step explanation:

The complete answers in the attached figure because is too long  

Part 1) Subtract (enter answer is standard form)

we have

[tex](-2x^4-2x^3+8x^2+2)-(x^4-5x^3+2x+7)[/tex]

Eliminate parenthesis

[tex]-2x^4-2x^3+8x^2+2-x^4+5x^3-2x-7[/tex]

Combine like terms

[tex](-2x^4-x^4)+(-2x^3+5x^3)+8x^2-2x+(2-7)[/tex]

[tex]-3x^4+3x^3+8x^2-2x-5[/tex] ------> standard form

Part 2) Which graph represents the inequality?

we have

[tex]2x-3y\leq 6[/tex]

Rewrite the inequality ------> [tex]3y\geq2x-6[/tex]

the solution is the shaded area above the solid line

the equation of the line is equal to [tex]3y=2x-6[/tex]

the slope of the line is positive

the y-intercept of the line is the point [tex](0,-2)[/tex] ---> value of y when the value of x is equal to zero

the x-intercept of the line is the point [tex](3,0)[/tex] ---> value of x when the value of y is equal to zero

The solution in the attached figure

Part 3) Multiply

we have

[tex]8y^4(2y^4-3y^3+5y)\\=( 8y^4*2y^4)-(8y^4*3y^3)+(8y^4*5y)\\=16y^8-24y^7+40y^5[/tex]

Part 4) Solve for x

we have

[tex]0=7x^2+x+5[/tex]

we know that

The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to

[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]

in this problem we have

[tex]7x^2+x+5=0[/tex]

so

[tex]a=7\\b=1\\c=5[/tex]

substitute in the formula

[tex]x=\frac{-1(+/-)\sqrt{1^{2}-4(7)(5)}} {2(7)}[/tex]  

[tex]x=\frac{-1(+/-)\sqrt{-139}} {14}[/tex]

remember that

[tex]i=\sqrt{-1}[/tex]

so

[tex]x=\frac{-1(+/-)\sqrt{139}i} {14}[/tex]

[tex]x1=\frac{-1+\sqrt{139}i} {14}[/tex]

[tex]x2=\frac{-1-\sqrt{139}i} {14}[/tex]

Part 5) Harold and Helen both spend [tex]3[/tex] hours reading a novel. Helen reads [tex]30[/tex] more pages than Harold. Helen reads at least [tex]40[/tex] pages in [tex]1[/tex] hour, but no more than [tex]70[/tex]pages in  [tex]1[/tex] hour.  

Which statement correctly describes the range of pages Harold reads?

Let

x------> the number of pages that Harold read

y------> the number of pages that Helen read

we know that

[tex]y=30+x[/tex] -----> equation A

[tex]y\geq 40\ pages[/tex] ------> in one hour

so

in three hours -------> [tex]y\geq 120\ pages[/tex] ------> inequality B

[tex]y\leq 70\ pages[/tex] ------> in one hour

so

in three hours ------->[tex]y\leq 210\ pages[/tex] ------> inequality C

Substitute equation A in the inequality B

[tex]x+30\geq 120\ pages[/tex] ------>  [tex]x\geq 90\ pages[/tex]

Substitute equation A in the inequality C

[tex]x+30\leq 210\ pages[/tex] ------> [tex]x\leq 180\ pages[/tex]

therefore  

Harold reads more than [tex]90[/tex] pages, but less than [tex]180[/tex] page

Part 6) Let

x------>  the number of Standard Specials sold last Monday  

y------>  the number of Deluxe specials sold last Monday

we know that

[tex]2x+4y\geq 200[/tex] -----> inequality A

[tex]0.50x+1.25y\leq100[/tex] -----> inequality B

[tex]x\geq30[/tex] -----> inequality C

Remember that

If a ordered pair representing a combination  of Standard specials and Deluxe specials could have been sold last Monday and make sense in the context of the situation

then

the ordered pair must satisfy all restrictions

Verify

A) [tex](30,68)[/tex]  

Substitute the value of x and y in each inequality

Inequality A

[tex]2(30)+4(68)\geq 200[/tex]

[tex]332\geq 200[/tex] ------> is true

Inequality B

[tex]0.50(30)+1.25(68)\leq100[/tex]

[tex]100\leq100[/tex] -----> is true

Inequality C

[tex]30\geq30[/tex] -----> is true

The ordered pair [tex](30,68)[/tex]  represent a combination of Standard specials and Deluxe specials could have been sold last Monday and make sense in the context of the situation

B) [tex](80,40.5)[/tex]  

The value of [tex]y=40.5[/tex] not make sense in the context of the situation, because is not a whole number

C) [tex](60,70)[/tex]  

Substitute the value of x and y in each inequality

Inequality A

[tex]2(60)+4(70)\geq 200[/tex]

[tex]400  \geq 200[/tex] ------> is true

Inequality B

[tex]0.50(60)+1.25(70)\leq100[/tex]

[tex]117.5\leq100[/tex] -----> is not true

therefore

the ordered pair [tex](60,70)[/tex] is not a solution

D) [tex](40,60)[/tex]  

Substitute the value of x and y in each inequality

Inequality A

[tex]2(40)+4(60)\geq 200[/tex]

[tex]320\geq 200[/tex] ------> is true

Inequality B

[tex]0.50(40)+1.25(60)\leq100[/tex]

[tex]95\leq100[/tex] -----> is true

Inequality C

[tex]40\geq30[/tex] -----> is true

The ordered pair[tex](40,60)[/tex]   represent a combination of Standard specials and Deluxe specials could have been sold last Monday and make sense in the context of the situation

E) [tex](50.5,40)[/tex]  

The value of [tex]x=50.5[/tex] not make sense in the context of the situation, because is not a whole number

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