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....true for all real numbers

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....Because the numerator (top part) is larger than the denominator, you know it will be more than 100%.
let percent be %; Since percentages are really just fractions with the denominator of , we can say
so, we can write
.......turn the denominator from a to a



.....since denominators same, nominators must be same too; so,

%

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or



(,)

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_____________


_______________

Solved by pluggable solver: Solve the System of Equations by Graphing

Start with the given system of equations: In order to graph these equations, we need to solve for y for each equation. So let's solve for y on the first equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets graph (note: if you need help with graphing, check out this solver) Graph of So let's solve for y on the second equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets add the graph of to our first plot to get: Graph of (red) and (green) From the graph, we can see that the two lines are parallel and will never intersect. So there are no solutions and the system is inconsistent.

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Solved by pluggable solver: Plot Any Inequality

Graphing function :

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Solved by pluggable solver: Solving a linear system of equations by subsitution

Lets start with the given system of linear equations Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y. Solve for y for the first equation Add to both sides Divide both sides by 6. Which breaks down and reduces to Now we've fully isolated y Since y equals we can substitute the expression into y of the 2nd equation. This will eliminate y so we can solve for x. Replace y with . Since this eliminates y, we can now solve for x. Distribute -2 to Multiply Reduce any fractions Add to both sides Combine the terms on the right side Now combine the terms on the left side. Multiply both sides by . This will cancel out and isolate x So when we multiply and (and simplify) we get <---------------------------------One answer Now that we know that , lets substitute that in for x to solve for y Plug in into the 2nd equation Multiply Add to both sides Combine the terms on the right side Multiply both sides by . This will cancel out -2 on the left side. Multiply the terms on the right side Reduce So this is the other answer <---------------------------------Other answer So our solution is and which can also look like (,) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) and (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle. and we can see that the two equations intersect at (,). This verifies our answer. ----------------------------------------------------------------------------------------------- Check: Plug in (,) into the system of equations Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Let and . Now plug those values into the equation Plug in and Multiply Add Reduce. Since this equation is true the solution works. So the solution (,) satisfies Since the solution (,) satisfies the system of equations this verifies our answer.

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area of a triangle is:


if given: and }
....plug in given values







if given: and
....plug in given values

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Use a protractor and a straightedge to draw a triangle that has a right angle and a 30 degree angle. Then measure the shortest and longest sides of the triangle to the nearest millimeter. What is the relationship of the two measurements.
1.
first draw a line segment and mark one point with C where you will have the vertex of 90° angle, you can use your protractor to draw this line, as the protractor includes a straight edge

2.
align the the guide markings on the protractor so that the line is at the 0-degree mark.

3.
make a mark by the 90° point on the protractor using a pen or pencil
4.
move the protractor so that the straight edge connects the vertex point and the mark you made at 90 degrees, use a straight edge of the protractor and connect the two points
5.
ones you got 90° , use one leg of the angle set a point B (the vertex point of 30° angle)to any convenient width, align the the guide markings on the protractor so that the line is at the 0-degree mark at B
6.
make a mark by the 30° point on the protractor
7.
use a straight edge of the protractor, aline point B and mark of the 30° point and draw a line from B through mark of the 30° point and continue until line intersect leg of the angle of 90° and you will get your right angle triangle
since one angle is 90° and another one is 30° , means that third angle is 60°
so, you will have 30° - 60° - 90° right triangle which and the ratio of this triangle's longest side to its shortest side is "two to one" which means, the longest side is twice as long as the shortest side

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first find equation of the line containing the pair of points
(,) and (,)

Solved by pluggable solver: FIND EQUATION of straight line given 2 points

hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (3, 0) and (x2, y2) = (0, -5). Slope a is . Intercept is found from equation , or . From that, intercept b is , or . y=(1.66666666666667)x + (-5) Your graph:

so, the slope is

or, find the slope this way:

Solved by pluggable solver: Finding the slope

Slope of the line through the points (3, 0) and (0, -5) Answer: Slope is

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The volume of a box is where , and
given:
the volume of a box is
the width of the box is ,
and it's height is more than it's length ; so,

to find: the height of the box
start with ...plug in given values




.......use quadratic formula











..we will find only positive solution because the length could be only positive number





now we can find the height

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...........add to both sides





..........add to both sides






so, is element of the set { ,,,.....}
or interval notation:
[,)

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A triangle is a right triangle whose internal angles are , and .
The three sides of a triangle have the following characteristics:
All three sides have lengths.
The shorter leg,, is half the length of the hypotenuse, . That is,

The longer leg's length, , is the shorter leg times . That is,

so, you are given
then, you have the shorter leg ...=>......=>...
and the longer leg's length is ..=>......=>...
check result using Pythagorean theorem:



...round decimal to whole number

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you are given:
if
if
if

this is a line parallel to and has at (,)
since you are given that you can find one more point choosing less than and draw that line all way from certain value of - to than stop; that will be at point (,
take second line and choose two values for so that
since this line is connected to the first line, use the point (,
and find one more point using



the second point is (,)
plot these points and connect them with a line segment

third part is or if
here you start with a point (,) last one you got and find one more point using any
let



the point is (,)
plot this point and draw a line from a point (,) and passes through (,)




from this graph where you see all three lines you need to take parts: from a line (red) draw part all way from the left to the point (,),
then got to second line (green) and draw a line segment which connects points (,) and (,),
then go to third line (blue) and draw from the point (,) all way down to the right
that will be your solution, the piecewise function

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you are given three roots, means a polynomial equation of least degree will be a polynomial of degree

where , , and


.....multiply
...recall

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if you have this than you have complex number and only what you can do is write real part as decimal number
but, if you have this than you multiply both nominator and denominator by to eliminate from denominator

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=

=

=

=

=

=

=

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...in base will be

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The of a polynomial are where the polynomial intersects the on the real coordinate plane. Mathematically speaking, these only occur when is equal to .
Polynomials can have because of the way they curve. The number of a certain polynomial can have is the of the polynomial.
A polynomial can only have . A degree can have to , but it doesn't have to have four. For even degree polynomials, it is possible that there are no .
degree polynomials must have .
The are key to graphing a polynomial. They are points that you can connect that lie on the . These are also known as to the .
So, how do we find these ? Simply, these points are where . So, you simply solve the polynomial for when or is .

Usually, cross the straight through. However, there is more than one way that the polynomial can intercept the . There are actually that the graph intercepts the . In the , it passes straight through no problem. In the , it goes down and touches the and then rebounds off it. In the , the graph sort of lingers around the interception point before crossing.
Why are there three types of intercepts? This is governed by a mathematical thing called . Multiplicity is the number of times a particular or solution appears. What if you ended up with the same twice? That means that that particular has a multiplicity of. It twice, and it therefore has a multiplicity of . An that times has a of .
Let us look at the following example:

We could commence the normal procedure for finding by setting the equal to and solving. However, we end up with twice, which means we get the intercept, (,), twice. The , (,) has a multiplicity of .



here are three cases how graph might cross
1. Normally, an has a multiplicity of , or it . When this happens, the graph simply passes straight through the . It occurs once, so it passes through and continues along with the normal path that it takes.
2. If the intercept has an multiplicity, meaning it occurs , times, times, etc., then the graph appears to touch the x-axis and then bounces off in the same direction it came from. The graph never passes through the x-axis, it simply touches it and goes back. As multiplicity increases, the valley will become flatter and flatter.
3.If the has an multiplicity, meaning it , times, etc., then the graph kind of lingers around the interception point before passing through. The graph does actually pass through, but it is sort of delayed before actually passing through, like in the image. As multiplicity increases, the deflection becomes closer and closer to the .

The graph of a polynomials of degree , for some constant is a line or is a line . It has no turning points and its tails are flat.
The graph of a polynomial of degree , ,with or is a slant line with , points, and tails in opposite direction.
The graph of a polynomial of degree . with (leading coefficient) is a parabola that opens if and if . The graph has turning point. It can have , , or , those that are obtained by shifts, stretching or shrinking and reflections in the negative (or negative) axis from . The function that is not shifted, like , have graphs with tails in opposite directions, one and no turning points. They are if and if .

-relative or minimum are values where the curve

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three geometric means between and :







is the ratio, so you multiply by (which equals ), giving you the of the three geometric means, multiply by (which equals ) to get the of the three geometric means, and multiply by (which equals ) to get the of the three geometric means
Final answer: ,,

Multiplying by gives you , so it is correct.

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two geometric means between and :









is the ratio, so you multiply by (which equals ), giving you the of the two geometric means, and multiply by (which equals ) to get the of the two geometric means

Final answer: ,

Multiplying by gives you , so it is correct.

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first find the equation of the line passing through (,) and (,)

Solved by pluggable solver: FIND EQUATION of straight line given 2 points

hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (-5, 2) and (x2, y2) = (-2, 5). Slope a is . Intercept is found from equation , or . From that, intercept b is , or . y=(1)x + (7) Your graph:

than find the equation of the line passing through (,) and (,)

Solved by pluggable solver: FIND EQUATION of straight line given 2 points

hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (-5, 2) and (x2, y2) = (5, -2). Slope a is . Intercept is found from equation , or . From that, intercept b is , or . y=(-0.4)x + (0) Your graph:

than find the equation of the line passing through (,) and (,)

Solved by pluggable solver: FIND EQUATION of straight line given 2 points

hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (-2, 5) and (x2, y2) = (5, -2). Slope a is . Intercept is found from equation , or . From that, intercept b is , or . y=(-1)x + (3) Your graph:

now graph all three lines together:
.........a slope is
.........a slope is

.........a slope is which is negative reciprocal of ; ...so, the line and are perpendicular to each other
if two of lines perpendicular to each other, and three intersecting line form a triangle, triangle is a right angle triangle

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....common denominator for and

....since , we will have






.....cross multiply




















solutions:
if ....=>...
if ....=>...
if , ...we need to use quadratic formula to find zeros










solutions:



and




so, your solutions are:

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..... write as

....group first three terms together
....factor out
....note that