Given the values of the linear functions f (x) and g(x) in the tables, where is (f – g)(x) positive?

A proportion maintains a consistent ratio m/d

If m = 0.75d, then the m/d ratio translates to (0.75d)/d = 0.75

In this case, the ratio can be expressed as (.75d-2)/d = 0.75 - (2/d).

Thus, it is not proportional.

Answer:

Option 1 is valid, which entails 2 hours of walking and 12 hours of running.

Step-by-step explanation:

The equations provided are:

3w + 6r ≥ 36

3w + 6r ≤ 90

We'll assess which options comply with these equations.

1) 2 hours walking; 12 hours running

w = 2 and r = 12

3w + 6r ≥ 36

3(2) + 6(12) ≥ 36

6+72 ≥ 36

78 ≥ 36

3w + 6r ≤ 90

3(2) + 6(12) ≤ 90

6+72 ≤ 90

78 ≤ 90

Both equations are satisfied. Option 1 is valid.

2) 4 hours walking; 3 hours running

w = 4 and r = 3

3w + 6r ≥ 36

3(4) + 6(3) ≥ 36

12+18 ≥ 36

30 ≥ 36 (this does not hold since 30 < 36)

3w + 6r ≤ 90

3(4) + 6(3) ≤ 90

12+18 ≤ 90

30 ≤ 90

Thus, Option 2 is invalid.

3) 9 hours running; 12 hours walking

w = 9 and r = 12

3w + 6r ≥ 36

3(9) + 6(12) ≥ 36

27+72 ≥ 36

99 ≥ 36

3w + 6r ≤ 90

3(9) + 6(12) ≤ 90

27+72 ≤ 90

99 ≤ 90 (this does not hold since 99 > 90)

Option 3 is invalid.

4) 12 hours walking; 10 hours running

w = 12 and r = 10

3w + 6r ≥ 36

3(12) + 6(10) ≥ 36

36+60 ≥ 36

96 ≥ 36

3w + 6r ≤ 90

3(12) + 6(10) ≤ 90

36 + 60 ≤ 90

96 ≤ 90 (this does not hold since 96 > 90)

So, Option 4 is invalid.

Response:

Second option:

Third option:

Detailed explanation:

The missing graph has been provided.

The attached image illustrates the graphing of the following system of linear equations:

Notice the intersection of the lines.

According to the definition, if lines in a system of equations intersect, then there is only one solution. This implies that the intersection point is the solution to that system. This can be expressed as:

Represented by "x" for the x-coordinate and "y" for the y-coordinate.

Here, it's noticeable that:

- The x-coordinate of the intersection point lies between   and .

- The y-coordinate of the intersection point is situated between   and .

Therefore, you can conclude that the forthcoming points (Refer to the options given in the exercise) are potential approximations for this system:

Solution

To find the angle m∠p.

Method of proof

In triangle ΔDAB, which is a right triangle

Applying the Pythagorean theorem gives us

Hypotenuse² = Perpendicular² + Base²

DB² = AB² + AD²

Where AB = 5 units

AD = 6 units

Substituting in the formula outcomes in

DB² = 5² + 6²

      = 25 + 36

     = 61

           = approximately 7.8 units

The triangle ΔDCB is also right-angled.

Using the trigonometric identity here.

Given that DC = 4 units and DB ≈ 7.8 units,

Substituting these values into the trigonometric identity gives us.

Thus, we find that ∠p ≈ 59.15 °

Response:

∠PQL=∠TRN [Angles corresponding]

Thus, PQ║RS and PQ=RS

Detailed explanation:

The side PQ has been drawn.

A second side QR is traced, forming an acute angle with side PQ.

Now side QR is extended to the left.

Create an arc from point Q such that it intersects QP at M and extends RQ at L. Without altering the compass width (i.e., the distance between the nib and pencil), draw an arc from R to intersect RQ at N. Now measure the distance LM with a compass. Position the compass at N and mark an arc cut from point R. Designate this intersection as T. Draw a line from point R through T. Then measure the length of PQ with the compass. Position your compass at R and create an arc on the produced line RT at S. Thus, we ascertain that PQ║RS and PQ=RS.

This occurs because

∠MQL=∠NRT [corresponding angles, with QR acting as the transversal]

∵PQ║RS  and PQ=RS [This identifies PQRS as a parallelogram]

Out of the four students who illustrated their explanations

Student 2 presented a partially correct but valid explanation.