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2 years ago

Directions: Follow the instructions for the following inequalities.

Mathematics

1 answer:

dmitriy555 [2]2 years ago

6 0

Answer:

1. 5050 < 8820

2. 15 > 6

3. -20 , -18

4. -1 < 1

Step-by-step explanation:

1. 4 < 7

multiply both sides by 7: 4 x 7 < 7 x 7 = 28 < 49

then multiply both sides by 6: 28 x 6 < 49 x 6 = 168 < 294

then multiply both sides by 3: 168 x 3 < 294 x 3 = 505 < 882

then multiply both sides by 10: 505 x 10 < 882 x 10 = 5050 < 8820

Did not affect the inequality sign

2. 11 > 2

Add 5 to both sides: 11 + 5 > 2 + 5 = 16 > 7

then add 3 to both sides: 16 + 3 > 7 + 3 = 19 > 10

then add -4 to both sides: 19 - 4 > 10 - 4 = 15 > 6

Did not affect the inequality sign

3. -4 < -2

Subtract 6 from both sides: - 4 - 6 < - 2 - 6 = -10 < -8

then subtract 8 from both sides: -10 - 8 < -8 - 8 = -18 < -16

then subtract 2 from both sides: -18 - 2 < -16 - 2 = -20 , -18

Did not affect the inequality sign

4. -8 < 8

Divide both sides by -4: -8 ÷ -4 < 8 ÷ -4 = 2 > -2

then divide both sides by -2: 2 ÷ -2 > -2 ÷ -2 = -1 < 1

Did affect the inequality sign throughout the operations, as when dividing by a negative number, flip the sign

Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged.
Multiplying or dividing both sides by a positive number leaves the inequality symbol unchanged.
Multiplying or dividing both sides by a negative number reverses the inequality. This means < changes to > (and vice versa)

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The standard form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .

<h3>How to find the standard equation of a hyperbola</h3>

In this problem we must determine the equation of the hyperbola in its standard form from the coordinates of the foci and a general equation of a circle. Based on the location of the foci, we see that the axis of symmetry of the hyperbola is parallel to the x-axis. Besides, the center of the hyperbola is the midpoint of the line segment with the foci as endpoints:

(h, k) = 0.5 · (- 1, 2) + 0.5 · (5, 2)

(h, k) = (2, 2)

To determine whether it is possible that the vertices are endpoints of the diameter of the circle, we proceed to modify the general equation of the circle into its standard form.

If the vertices of the hyperbola are endpoints of the diameter of the circle, then the center of the circle must be the midpoint of the line segment. By algebra we find that:

x² + y² - 4 · x - 4 · y + 4= 0

(x² - 4 · x + 4) + (y² - 4 · y + 4) = 4

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The center of the circle is the midpoint of the line segment. Now we proceed to determine the vertices of the hyperbola:

V₁(x, y) = (0, 2), V₂(x, y) = (4, 2)

And the distance from the center to any of the vertices is 2 (semi-major distance, a) and the semi-minor distance is:

b = √(c² - a²)

b = √(3² - 2²)

b = √5

Therefore, the standard form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .

To learn more on hyperbolae: brainly.com/question/27799190

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