i need help with the below question please

Question

Part 1

The equation 4x+1=3x−7 is of the form f(x)=g(x). Complete parts a​, b​, and c.

a. Solve the equation analytically.

b. Solve

f(x)>g(x).

c. Solve

f(x)<g(x).

Question content area bottom

Part 1

a. Select the correct answer below​ and, if​ necessary, fill in the answer box within your choice.

A.

The solution set is the finite set of numbers

​(Type integers or simplified fractions. Use a comma to separate answers as​ needed.)

B.

The solution set is the interval

enter your response here.

​(Type integers or simplified fractions. Type your answer in interval​ notation.)

C.

The solution set is ∅.



Part 2

b. Select the correct answer below​ and, if​ necessary, fill in the answer box within your choice.

A.

The solution set is the finite set of numbers enter your response here.

​(Type integers or simplified fractions. Use a comma to separate answers as​ needed.)

B.

The solution set is the interval

​(Type integers or simplified fractions. Type your answer in interval​ notation.)

C.

The solution set is ∅.

Part 3

c. Select the correct answer below​ and, if​ necessary, fill in the answer box within your choice.

A.

The solution set is the finite set of numbers

​(Type integers or simplified fractions. Use a comma to separate answers as​ needed.)

B.

The solution set is the interval

​(Type integers or simplified fractions. Type your answer in interval​ notation.)

C.

The solution set is ∅

Answer 1

You only show a Part 1, but I see answers available for Part 2 and 3.

I am assuming a, b and c are mean to be part 1, part 2 and part 3 in this answer. Additionally I am assuming x is allowed to be any real number.

So we know

f(x) = 4x+1, g(x) = 3x-7

If (Part 1) f(x) = g(x), then

4x+1 = 3x -7 which means that (subtracting 3x from both sides)

x+1 = -7 which means that (subtracting 1 from both sides)

x = -8.

Note that x is a single value, which is a finite set containing 1 element.

Part 2 f(x) > g(x)

This means that

4x+1 > 3x -7

and we can subtract 3x from both sides

x+1> -7 and subtracting 1 from both sides

x > -8.

Note that -7.1234 is greater than -8, so is 0, so is pi, so is sqrt(2) and so on, there are an infinite number of numbers greater than -8, so the solution set is infinite, and the interval for solutions is (-8,infinity) in interval notation

Finally

f(x) < g(x)

means that

4x+1 < 3x -7

subtracting 3x from both sides

x+1 < -7

subtracting 1 from both sides

x < -8

note that -9, -5pi , -sqrt(134) and so on are all less than -8, so again the solution set is an infinite set of numbers, with x< -8

so in interval notation the solution set is (-infinity, -8)

Answer 2

x=-8 when f(x)=g(x)

4(-8)+1= 3(-8)-7

-32+1= -24-7

-31=-31

4x+1=3x-7

4x-3x=-7-1

x =-8

solution set = {-8}

x>-8 when f(x)>g(x)

x is an element of the interval (-8, infinity)

x<-8 when f(x)<g(x)

x is an element of the interval (-infinity, -8)