QUICK PLEASE I NEED HELP WITH THIS GEOMETRY QUESTION!
I dont know if what I selected is right

9514 1404 393

  your choice is the best

Step-by-step explanation:

If you simply look at the segments marked with expressions in z, you have* ...

  z+23 > 0   ⇒   z > -23

  3z +9 > 0   ⇒   z > -9/3 = -3

That is, the side lengths must be positive. We must also have the side labeled z+23 be longer than the side labeled 3z+9:

  z +23 > 3z +9

  14 > 2z . . . . . . . subtract z+9

  7 > z

So, the limits on the marked side lengths require ...

  -3 < z < 7 . . . . matches the 4th choice

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If you actually work out the possible sizes of these triangles such that the angles are the given measures, you find that the minimum length of the legs with the hash mark is about 15.8649, and the minimum value of z is ...

  z > (12√6 -7)/5 ≈ 4.47878

This value is obtained when the triangles are isosceles. Apparently, you're not supposed to think that deeply about this question.

* The "or equal to" case is not included, because that would allow the triangles to have zero area. That is, the triangles would be a single point.

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is