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Sophie D.

Union

union means OR and you shade AWAY (obviously) from the points. You write as two inequalities (because they are not linked by any common answers.)

Writtien as U

Intersection

intersection means AND. you shade BETWEEN the points. you write as one inequality.

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Critical points

critical points are the values of x that make the polynomial equal to zero. these zeros or critical points divide the number line into... TEST INTERVALS.

test intervals

where the polynomial is either positive or negative. you use critical points to create test intervals.

types of inequalities

1. rational inequalitities (a fraction)

2. quadratic inequalities (factor ect...)

3. linear inequalities (one that is not quadraic or rational ie; fraction)

4. absolute value inequalities

quadratic inequalities

1. you bring the whole thing to one side. 2. you factor it or do another method of solving quadratic ex: sq. root meth. (while it is set equal to zero) 3. the 2 answers you get are the 2 critical points. 4. you test it out and see what works. 5. the you see if its union, intersection etc.

Rational inequalities

1. you follow the same steps except as in quadratic inequalities except in rational ineq. the critical points are any values that make either the num. or denom. equal to zero. (this is why all the long you are setting things equal to zero- to find the critical points.) 2. finish solving it/ writing it out the same way.

linear inequalities

1. you solve like you would a normal equation 2. you write it in interval notation based on what it is saying.

absolute value inequalities

1. you set the stuff inside absolute value equal to the negative amount on the other side (in this case you flip the sign) and you set it equal normally.

2. you finish simplifying. you get a range for x.

3. you finish writing it in interval notation/ on a number line.

4. you don't find critical points/test intervals with absolute value.

completing the square

1. make the lead coefficient 1 by dividing the equation by a. 2. subtract c from both sides. 3. Square half of b and add this to both sides. 4. write left of eq. as a perfect square 5. you can now solve with the Sq. root method.

Nature of the roots and how they relate to a graph

discriminant: b sq. -4ac

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b sq. -4ac =negative number

-you have 2 imaginary solutions (complex conjugates)

-the graph is a U with no x-intercepts

b sq. -4ac is equal to zero

-you have 1 real solution counted twice (double root)

-the graph is a U with 1 x-intercept

b sq. -4ac is a positive number

It can be 1. perfect square 2. positive non-perfect square

For both you get two real distict solutions. But the perfect sq. is rational while the non-perfect sq. is irrational. The first one is also factorable.

-When you graph it, you get 2 x-intercepts.

GreatOR

if the part with x is greatOR than the thing on the other side, you know to write it as two inequalities, you shade away, and its a union.

less thAND

if the part with the x is less thAND the thing on the other side, you know to write it as one inequality (it can be, you don't have to) , you shade between, and its an intersection.

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