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konstantin123 [22]
3 years ago
12

PLS HELP LAST DAY OF APEX a.graph this system of inequalities y<-x+4 y>-x-2 b. How many solutions does this system of ineq

ualities have?

Mathematics
1 answer:
Ede4ka [16]3 years ago
4 0

Answer:

Part ) The graph in the attached figure

Part b) The system has infinitely solutions

Step-by-step explanation:

Part a) Graph this system of inequalities

we have

y < -x+4 -----> inequality A

The solution of the inequality A is the shaded area below the dotted line

y=-x+4

The y-intercept of the dotted line is the point (0,4)

The x-intercept of the dotted line is the point (4,0)

The slope of the dotted line is negative (m=-1)

y > -x-2 -----> inequality B

The solution of the inequality B is the shaded area above the dotted line

y=-x-2

The y-intercept of the dotted line is the point (0,-2)

The x-intercept of the dotted line is the point (-2,0)

The slope of the dotted line is negative (m=-1)

using a graphing tool

The graph in the attached figure

Part b)  How many solutions does this system of inequalities have?

we know that

The solution of the system of inequalities is the shaded area between the two dotted lines (see the attached figure)

therefore

The system has infinitely solutions

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The function which is same as the function y = 3cos(2(x +π/2)) -2 is: Option A: y= 3sin(2(x + π/4)) - 2

<h3>How to convert sine of an angle to some angle of cosine?</h3>

We can use the fact that:

\sin(\theta) = \cos(\pi/2 - \theta)\\\sin(\theta + \pi/2) = -\cos(\theta)\\\cos(\theta + \pi/2) = \sin(\theta)

to convert the sine to cosine.

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(this all positive negative refers to the fact that if you use given angle as input to these functions, then what sign will these functions will evaluate based on in which quadrant does the given angle lies.)

Here, the given function is:

y= 3\cos(2(x + \pi/2)) - 2

The options are:

  1. y= 3\sin(2(x + \pi/4)) - 2
  2. y= -3\sin(2(x + \pi/4)) - 2
  3. y= 3\cos(2(x + \pi/4)) - 2
  4. y= -3\cos(2(x + \pi/2)) - 2

Checking all the options one by one:

  • Option 1: y= 3\sin(2(x + \pi/4)) - 2

y= 3\sin(2(x + \pi/4)) - 2\\y= 3\sin (2x + \pi/2) -2\\y = -3\cos(2x) -2\\y = 3\cos(2x + \pi) -2\\y = 3\cos(2(x+ \pi/2)) -2

(the last second step was the use of the fact that cos flips its sign after pi radian increment in its input)
Thus, this option is same as the given function.

  • Option 2: y= -3\sin(2(x + \pi/4)) - 2

This option if would be true, then from option 1 and this option, we'd get:
-3\sin(2(x + \pi/4)) - 2= -3\sin(2(x + \pi/4)) - 2\\2(3\sin(2(x + \pi/4))) = 0\\\sin(2(x + \pi/4) = 0

which isn't true for all values of x.

Thus, this option is not same as the given function.

  • Option 3: y= 3\cos(2(x + \pi/4)) - 2

The given function is y= 3\cos(2(x + \pi/2)) - 2 = 3\cos(2x + \pi) -2 = -3\cos(2x) -2

This option's function simplifies as:

y= 3\cos(2(x + \pi/4)) - 2 = 3\cos(2x + \pi/2) -2 = -3\sin(2x) - 2

Thus, this option isn't true since \sin(2x) \neq \cos(2x) always (they are equal for some values of x but not for all).

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Thus, this function is not same as the given function.

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