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Featured 3 months ago

**Given:**

We are given the **System of Inequalities**:

**and**

**graph created using GeoGebra**

**graph created using GeoGebra**

**and** **combined graphs created using GeoGebra**

If you observe closely, you will find the **solution** in a visual form.

The **solution to the system of inequalities** is **the darker shaded region, which is the overlap of the two individual regions**.

If you want to view just the solutions, please refer to the image below:

**Note:**

If the inequality is < or >, graph of the equation has a **dotted line**.

If the inequality is ≤ or ≥, graph of the equation has a **solid line. **

This line **divides the xy- plane into two regions:** a region that **satisfies the inequality**, and **a region that does not.**

Featured 2 months ago

speed = distance / time

here,

and

she ran

when she ran for

this gives the two equations,

you can solve both by isolating distance

this means that

using this, you can solve for time

then

the distance of the race is

Featured 2 months ago

Let,the coordinate of

So,if

Now,midpoint of

clearly,this point will lie on

So,

or,

And this will lie as well on

so,

or,

So,the coordinate is

Featured 2 months ago

#"the initial statement is "ypropx/z^2#

#"to convert to an equation multiply by k the constant"#

#"of variation"#

#rArry=kxxx/z^2=(kx)/z^2#

#"to find k use the given condition"#

#y=1/6" when "x=20" and "z=6#

#y=(kx)/z^2rArrk=(yz^2)/x=(1/6xx36)/20=3/10#

#"equation is " color(red)(bar(ul(|color(white)(2/2)color(black)(y=(3x)/(10z^2))color(white)(2/2)|)))#

#"when "x=14" and "z=5" then"#

#y=(3xx14)/(10xx25)=21/125#

Featured 2 months ago

To simplify this expression, all we need to do, is use

First thing we need to know is the definition of

This is the order we must do this, from top to bottom. This process is more commonly known as the Order of Operations .

Now that we know the order of the steps we must take to solve this, this is the solution:

Featured 1 month ago

Given:

#x^3+y^3#

Note that if

#x^3+y^3=(x+y)(x^2-xy+y^2)#

We can calculate the discriminant for the remaining homogeneous quadratic in

#x^2-xy+y^2#

is in standard form:

#ax^2+bxy+cy^2#

with

This has discriminant

#Delta = b^2-4ac = color(blue)(1)^2-4(color(blue)(1))(color(blue)(1)) = -3#

Since

We can factor it with complex coefficients by completing the square and using

#x^2-xy+y^2 = (x-1/2y)^2+3/4y^2#

#color(white)(x^2-xy+y^2) = (x-1/2y)^2+(sqrt(3)/2 y)^2#

#color(white)(x^2-xy+y^2) = (x-1/2y)^2-(sqrt(3)/2 y i)^2#

#color(white)(x^2-xy+y^2) = ((x-1/2y)-sqrt(3)/2i y)((x-1/2y)+sqrt(3)/2i y)#

#color(white)(x^2-xy+y^2) = (x-(1/2+sqrt(3)/2i)y)(x-(1/2-sqrt(3)/2i)y)#

So:

#x^3+y^3 = (x+y)(x^2-xy+y^2)#

#color(white)(x^3+y^3) = (x+y)(x-(1/2+sqrt(3)/2i)y)(x-(1/2-sqrt(3)/2i)y)#

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