Ineed to know how to do this i don't understand

Okay so angles 1 and 2 in problem 1 are congruent because alternate interior angles between two parallel line and a transversal are congruent. Angles 1 and 2 in problem 2 are congruent because they are vertical angles and vertical angles are always congruent.
To find x in problem 3, you can set the expressions to equal each other because alternate exterior angles between two parallel lines and a transversal are congruent; it would look like this:
x + 24 = 3x - 8
x + 32 = 3x (add 8 to each side)
32 = 2x (subtract x from each side)
16 = x (divide each side by x)
For problem 3, x = 16.
To find x in problem 4, set the expressions plus each other to equal 180: same side interior angles between two parallel lines and a transversal are supplementary (they add to equal 180); it would look like this:
2/3(x + 27) + 3x - 25 = 180
2/3x + 18 + 3x - 25 = 180 (apply the distributive property)
3.667x - 7 = 180 (combine like terms)
3.667x = 187 (add 7 to both sides)
x = 50.995
We should be checking our work for these; let's do that:
For the first problem - 16 + 24 = 3(16) - 8  =>  40 = 48 - 8  <--This is true, so the first one is correct.
Now to check the second one:
0.667(50.995 + 27) + 3(50.995) - 25 = 180
34.013 + 18.009 + 152.985 - 25 = 180
180.007 = 180 <---This one checks out as well. x = 50.995 for problem 4.

Hope this helps.

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.