Prove or disprove each of these statements about the floor and ceiling functions. a) ⌊⌈x⌉⌋ = ⌈x⌉ for all real numbers x. b) ⌊x+y⌋=⌊x⌋+⌊y⌋for all real numbers x and y. c) ⌈⌈x∕2⌉ ∕2⌉ = ⌈x∕4⌉ for all real numbers x. d)⌊√⌈x⌉⌋ = ⌊√x ⌋ for all positive real numbers x. e) ⌊x⌋+⌊y⌋+⌊x+y⌋≤⌊2x⌋+⌊2y⌋forall real numbers x and y.

A) equality function will hold

B) equality function doesn't hold

C)  equality function holds )

D) equality function doesn't hold

E) equality function holds

Step-by-step explanation:

a) ⌊⌈x⌉⌋ = ⌈x⌉ for all real numbers x.

The ceiling function will yield an integer, that is greater ≥ x.

Also the floor function when applied to an integer will yield the same integer value. hence the two sides will become the same Therefore the  equality function will hold

b) ⌊x+y⌋=⌊x⌋+⌊y⌋ for all real numbers x and y

The floor function of this statement yields a value that is ≤ inserted value and this makes the right hand side of the statement to be 1 less than the sum produced from the left hand side of the statement, therefore the equality function doesn't hold

c) ⌈⌈x∕2⌉ ∕2⌉ = ⌈x∕4⌉ for all real numbers x.

The ceiling function produces an integer that is ≥ value inserted and the integer gotten is divided by 2 and this serves a s an input to the ceiling function( which means X has been divide by 4 ) making the two sides of the equation the same ( equality function holds )

d)⌊√⌈x⌉⌋ = ⌊√x ⌋ for all positive real numbers x

The ceiling function is applied to the value inserted in x(on the left hand side ), and the result will not be equal to that in the right hand side.

Therefore the equality function doesn't hold

e) ⌊x⌋+⌊y⌋+⌊x+y⌋≤⌊2x⌋+⌊2y⌋forall real numbers x and y.

If the value of the fractional part of x or y is > 0.5, then the right hand side will be greater  than the left hand side by 1.

Otherwise, both left and right hand sides yield equal answers. therefore equality function holds

segment bisector

step-by-step explanation:

line segment bisector. definition: a line, ray or segment which cuts another line segment into two equal parts. try this drag one of the orange dots at a or b and note the the line ab always divides the line pq into two equal parts.

mark brainliest!

5x + 3y = - 4

step-by-step explanation:

the equation of a line in standard form is

ax + by = c ( a is a positive integer and b, c are integers )

express the line first of all in slope- intercept form

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

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = (7, - 3) and (x₂, y₂ ) = (4, - 8)

m = = -

y = - x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, - 8 ), then

- 8 = - + c ⇒ c = -

y = - x - ← in slope-intercept form

multiply all terms by 3 to eliminate the fractions

3y = - 5x - 4 ( add 5x to both sides )

5x + 3y = - 4 ← in standard form