If f(x) = 4x + 1 and g(x) = x^2 - 5, find (f - g)(x)

Solution:

Given equation of line [tex]-4x+2y=24[/tex]

To find the x-intercept of the equation, substitute [tex]y=0[/tex] in the equation,

[tex]\Rightarrow -4x+2(0)=24\\\Rightarrow -4x=24\\\Rightarrow x=-\frac{24}{4} =-6[/tex]

Hence,  x-intercept of the equation is [tex](-6,0)[/tex].

To find the y-intercept of the equation, substitute [tex]x=0[/tex] in the equation,

[tex]\Rightarrow -4(0)+2y=24\\\Rightarrow 2y=24\\\Rightarrow y=\frac{24}{2} =12[/tex]

Hence,  y-intercept of the equation is [tex](0,12)[/tex].

the slope of a perpendicular line = - [tex]\frac{3}{5}[/tex]

the equation of a line in slope-intercept form is

y = mx + c ( m is the slope and c the y-intercept )

rearrange 5x - 3y = 2 into this form

subtract 5x from both sides

- 3y = - 5x + 2 ( divide all terms by - 3 )

y = [tex]\frac{5}{3}[/tex] x - [tex]\frac{2}{3}[/tex]

with slope m = [tex]\frac{5}{3}[/tex]

the slope of a line perpendicular = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{3}{5}[/tex]

Answer:

c

Step-by-step explanation:

The inequality equivalent to -m ≥ 15 is m ≤ -15, achieved by multiplying both sides of the inequality by -1, which reverses the inequality symbol. So the correct answer is C. m ≤ -15.

The question asks which inequality is equivalent to the inequality -m ">≥ 15. To find the equivalent inequality where m is positive, we need to multiply or divide both sides of the inequality by -1, remembering that doing so reverses the inequality symbol. Therefore, multiplying both sides by -1 gives m ≤ -15, which means that the inequality that is equivalent to -m ≥ 15 is m ≤ -15.

So the correct answer is C. m ≤ -15.

Answer:

The Average rate of the first plane is 560 mph

Step-by-step explanation:

x = first plane's average rate (in mph)

x−108 = second plane's average rate (in mph)

First planes total distance = 2.25x

Second planes total distance = 1.25(x−108)

2.25x + 1.25(x−108) = 1825

x = 560

Answer:

1 3/5

Step-by-step explanation:

4/15÷1/6

Copy dot flip

4/15 * 6/1

24/15

Divide the top and bottom by 3

8/5

Change the improper fraction to a mixed number

5 goes into 8 1 time with 3 left over

1 3/5

For this case we have the following expressions:

[tex]x = -4y + 4\\\\2x + 8y = 8\\[/tex]

Rewriting the second equation we have:

[tex]2x = -8y + 8\\[/tex]

It is observed that by multiplying the first equation by 2, the second equation is obtained.

Thus, the equations are linearly dependent and since the lines are equal then they are intercepted in infinite points. Therefore, the system has endless solutions.

Answer:

Option D

Answer:

6i\sqrt{19}

Step-by-step explanation:

- 9 ≤ x ≤ 7

the domain is the values of x on the x- axis that define y = f(x)

y is defined for all values of x in the stated domain

You would first substitute x and y into the equation so: -2(-2)^2+4(3)

Then you solve:

-2(-2)^2+4(3)

8+12

so your answer will be 20

Your identity says ...

... sum of cubes = (sum)³ -3(product)(sum)

... = 4³ -3·1·4

... = 64 -12 = 52

_____

The two numbers are 2±√3, and the sum of their cubes is indeed 52.

[tex]\frac{8}{27}[/tex]

(2/3)^3 = [tex]\frac{2^{3} }{3^{3} }[/tex] = (2 × 2 × 2 )/(3 × 3 × 3 ) = [tex]\frac{8}{27}[/tex]

Answer:

Step-by-step explanation:

The two short sides must be longer than those required in an isosceles right triangle. The ratio of side length to hypotenuse length in a 45°-45°-90° triangle is 1 : √2. So, such a triangle requires side lengths of 30/√2 ≈ 21.213 inches. Its perimeter will be about

... (30 +21.213 +21.213) in = 72.426 in

Since the perimeter needs to be slightly longer than that for the triangle to be acute, the smallest possible perimeter is ...

... 72.43 in

You want to get y by itself

y=1/2 x

y=mx+b

m=1/2 b=0

y = [tex]\frac{1}{2}[/tex] x + 0

given 2y = x ( divide both sides by 2 )

y = [tex]\frac{1}{2}[/tex] x + 0

Answer:

1) C

2) A

3) D

Step-by-step explanation:

A) We substitute the values for a ,b , c and d in the given equation.[tex]\frac{6}{3} +4^2 (5) \\= 2+80\\=82[/tex]

Hence option c. 82 is the answer.

2) The product 6(-3)

Since positive x negative gives negative

we get -(6x3) = -18

Option A is the answer.

3) For this we have to find cube of a fraction with numerator 4 and denominator 7

We find cube of numerator first = 4x4x4 =64

Cube of denominator = 7x7x7 = 343

Hence cube of fraction = 64/343

Option D

Answer:

Option C is right answer

Step-by-step explanation:

We have in coordinate geometry a straight line in slope intercept form as

y = mx+c

where m is the slope and c is the y intercept

There are special lines when m=0 and m = infinity.

When a line is parallel to y axis, it has slope = tan90 = infinity.

Hence line will have equation of the form x = a for some a.

Since the line passes through (3,2) we can find a using this.

Substitute x value to get 3 = x

Hence equation of the line is x =3

Answer:

X=3

Step-by-step explanation:

The function f(x) = e^x never crosses the x-axis and crosses the y-axis at point (0,1). It does not pass through the point (e,1), but it does pass through the point (1,e). Hence, the correct statement is 'It passes through the point (1,e)'.

The function f(x) = e^x is an exponential function where 'e' is Euler's number, approximately equal to 2.71828. Its graph has distinctive traits which are related to the options provided.

The statements 'It crosses the x-axis at e' and 'It crosses the y-axis at e' are incorrect because the graph of f(x) = e^x never crosses the x-axis, and it crosses the y-axis at the point (0,1), not e.

The statement 'It passes through the point (e,1)' is also incorrect. This function, f(x) = e^x, will pass through the point (1,e), not (e,1). Hence, the option 'It passes through the point (1,e)' is the true statement about the graph of the function.

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Answer:

2 1/10

Step-by-step explanation:

(5 1/4)/(2 1/2)

Change mixed fraction to improper fractions

(21/4)/(2 1/2)

(21/4)/(5/2) =21/10

=2 1/10

Answer:

2 1/10

Step-by-step explanation:

The equation of the circle in standard form with a center at (2, -3) and a radius [tex]\(4\sqrt{5}\)[/tex] is [tex]\( (x - 2)^2 + (y + 3)^2 = 80 \)[/tex].

To find the equation of the circle with the diameter endpoints given,

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

Midpoint [tex]\((M_x, M_y)[/tex] = [tex]\left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)\)[/tex]

For the endpoints A (-2,-5) and B (6,-1), we calculate:

[tex]\(M_x = \frac{-2 + 6}{2} = \frac{4}{2} = 2\)[/tex]
[tex]\(M_y = \frac{-5 + (-1)}{2} = \frac{-6}{2} = -3\)[/tex]

So, the midpoint, which is the center of the circle, is C (2, -3).

The radius is half the length of the diameter, and the length of the diameter is the distance between the endpoints A and B using the distance formula:

Distance d = [tex]\sqrt{(x2 - x1)^2 + (y2 - y1)^2} \)[/tex]

r = [tex]\frac{d}{2} = \frac{\sqrt{(6 - (-2))^2 + (-1 - (-5))^2}}{2} \)[/tex]

Calculating the expressions:

r = [tex]\frac{\sqrt{(6 + 2)^2 + (-1 + 5)^2}}{2} \)[/tex]
 = [tex]\frac{\sqrt{8^2 + 4^2}}{2} \)[/tex]
 = [tex]\frac{\sqrt{64 + 16}}{2} \)[/tex]
 = [tex]\frac{\sqrt{80}}{2} \)[/tex]
 = [tex]\frac{8\sqrt{5}}{2} \)[/tex]

r = [tex]4\sqrt{5} \)[/tex]

The equation of a circle with center (h, k) and radius r can be represented as:
[tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex]

Plugging our midpoint as the center and our radius into the equation:

[tex]\( (x - 2)^2 + (y + 3)^2 = (4\sqrt{5})^2 \)[/tex]

Simplifying,
[tex]\( (x - 2)^2 + (y + 3)^2 = 16 \cdot 5 \)[/tex]
[tex]\( (x - 2)^2 + (y + 3)^2 = 80 \)[/tex]

Thus, the required equation is [tex]\( (x - 2)^2 + (y + 3)^2 = 80[/tex].

Answer:

1) Volume of water displaced = 5.16 L

2) Copper sinks in water.

Explanation:

1)  Density of copper = 8.92[tex]g/cm^3[/tex]

 Mass of copper = 46 kg = 46000 g

 We know that, Density = Mass / volume

        So, Volume = Mass/ density

              Volume of copper = 46000/8.92 = 5156.95 [tex]cm^3[/tex]

  Since the density of copper 8.92[tex]g/cm^3[/tex] is greater than density of water 1[tex]g/cm^3[/tex], the copper immerses in water. Since it immerses it will displace a volume of water which is equal to the volume of copper.

  So volume of water displaced = volume of copper = 5156.95 [tex]cm^3[/tex] =5.16 L of water

2)  Since the density of copper 8.92[tex]g/cm^3[/tex] is greater than density of water 1[tex]g/cm^3[/tex], the copper immerses in water.  So copper sink in water.

Write and solve an inequality:

Total(t) = $5.42 + ($1.08/therm)(t) ≤ $87.50.

First, subtract $5.42 from $87.50:  then ($1.08/therm)(t) ≤ $82.08

Next, divide both sides by $1.08 per therm:

t ≤ ($82.08) / ($1.08/therm) = 76 therms (maximum).  Thus, Rameen's usage could be [0,76] (therms).

Answer:

[0,76]

Step-by-step explanation:

Step 1. Read the problem.

Step 2. Identify what you are looking for.

the number of therms Rameen can buy

Step 3. Name what you are looking for. Choose a variable to represent that quantity.

Let t= the number of therms.

Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality.

$5.42 plus $1.08 times the number of therms is less than or equal to $87.50.

5.42+1.08t≤87.50

Step 5. Solve the inequality.

1.08tt≤82.08≤76

Step 6. Check the answer in the problem and make sure it makes sense.

Yes, 5.42+1.08(76)=87.50.

Step 7. Answer the question with a complete sentence.

Rameen can buy no more than 76 therms if he wants his heating bill to be a maximum of $87.50.

Since it is not possible to buy a negative number of therms, the lest number of therms Rameen can buy is 0.

In interval notation, the amount of therms Rameen can buy is [0,76].

Answer:

[tex]1098[/tex] square feet of glass is needed to replace the window.

Step-by-step explanation:

Square feet is the unit of measure of Area.  How many square feet of glass are required to replace the window can be found by calculating the area of the window.

Area of the window = Width * Length

=[tex]\frac{122}{3}*\frac{81}{3}[/tex]

=[tex]\frac{122*81}{3*3}[/tex]

=[tex]\frac{9882}{9}[/tex]

=[tex]1098[/tex] square feet.

∴ we need [tex]1098[/tex] square feet of glass.

1098 ft²

Area = width × height

         = [tex]\frac{122}{3}[/tex] × [tex]\frac{81}{3}[/tex]

        = [tex]\frac{122(81)}{3(3)}[/tex] = [tex]\frac{9882}{9}[/tex]= 1098 ft²

Triangle ABC is equilateral, because AB=BC=AC=a. Then

m∠A=m∠B=m∠C=60°.

Let point D lie on the ray BC to the right from points B and C and let CD=x. Then BD=a+x, AB=a.

Consider triangle ACD. In this triangle, m∠ACD=180°-m∠ACB=180°-60°=120°.

By the cosine theorem,

[tex]AD^2=AC^2+CD^2-2\cdot AC\cdot CD\cdot \cos \angle ACD,\\\\AD^2=a^2+x^2-2\cdot a\cdot x\cdot \cos 120^{\circ},\\\\AD^2=a^2+x^2+ax,\\\\AD=\sqrt{a^2+x^2+ax}.[/tex]

Since [tex]a^2+x^2+ax=a^2+x^2+ax+ax-ax=(a+x)^2-ax,[/tex] then

[tex]AD^2=(a+x)^2-ax<(a+x)^2=BD^2\Rightarrow AD<BD[/tex]

and

[tex]AD^2=a^2+x^2+ax>a^2=AB^2\Rightarrow AD>AB.[/tex]

Therefore, you get double inequality

[tex]AB<AD<BD[/tex] or [tex]BD>AD>AB.[/tex]

Answer:

The proof is below

Step-by-step explanation:

Please look at the attachment below

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Mrs. Riley can make 168 pillows because she bought 168.75 yards of fabric and each pillow requires 1 yard of fabric.

First, we need to determine how many yards of fabric Mrs. Riley purchased. We do this by dividing the total cost of the fabric ($405) by the cost per yard ($2.40). So, 405 ÷ 2.4 = 168.75 yards. Each pillow requires 1 yard of fabric, therefore she can sew 168 pillows with no leftover fabric.

Here is the calculation in more detail:

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let h= history books

science books = s

h=2s  there are 2 times as many history as science so to get them equal we double the science

h+s=81    there are 81 total history and science books

2s+s=81   substitute 2s for h in the above equation

3s = 81   combine like terms

s = 27

There are 27 science books

h=2s

h=2(27)

h=54

there are 54 history books

[tex]|\Omega|=12^2=144\\|A|=4\cdot3=12\\\\P(A)=\dfrac{12}{144}=\dfrac{1}{12}[/tex]

Answer:

Step-by-step explanation:

subtracting term by term

(3.1x - 4.3x ) + (2.8z - (- 1.2z ) )

= - 1.2x + (2.8z + 1.2z ) = - 1.2x + 5z

The primes in the range 1–38 are ...

... 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

There are 12 of them, so the probability of drawing a prime is

... 12/38 = 6/19

Answer:

[tex]\text{Probability}=\frac{6}{19}[/tex]      

Step-by-step explanation:

Given : Balls numbered from 1 to 38 are placed in a container and stirred. If one is drawn at random.

To find : The probability that the number is a prime number

Solution :  

Probability is defined by,

[tex]\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total number of outcome}}[/tex]

Favorable outcomes - To get the number is prime number from 1 to 38

Prime numbers are those which were not divisble by any number except 1 and itself.

From 1 to 38 - 2,3,5,7,11,13,17,19,23,29,31,37

Favorable outcome = 12

Total number of outcome is 1 to 38 = 38

[tex]\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total number of outcome}}[/tex]

[tex]\text{Probability}=\frac{12}{38}[/tex]

[tex]\text{Probability}=\frac{6}{19}[/tex]                                          

if im not mastaken the answer is -48