# Review topics for final exam

**2.1 – one variable linear equations
**

solve equations (isolate x)

including those with fractions or decimals

**2.5 – one variable linear inequalities**

solve inequalities (isolate x; remember when to flip the symbol)

including those with fractions or decimals

**2.6 – set operations - union, intersection**

find intersection ( ∩ or “and”) of two sets

find union (∪ or “or”) of two sets

solve compound inequalities (with “and” or “or”)

**3.1 - rectangular coordinate system (x-y plane)**

quadrants

plot ordered pairs

graph any equation by using an x-y table

find x- and y- intercepts (from a graph or from an equation)

midpoint*

**3.2 and 3.3 – graphs of linear equations**

find slope from 2 points (given the graph or just ordered pairs)

don’t forget slopes for horizontal and vertical

graph lines

given the slope and any point on the line

given an equation

use different forms of linear equations - slope-intercept, point-slope

what important values are in these?

how/when do you use each?

horizontal lines (y = some #)

vertical lines (x = some #)

find equations of parallel & perpendicular lines and their slopes

given a graph, use one of the linear forms to write an equation of the line

remember for word problems: y = mx+b

m = rate of change (increase)

b = initial/starting value

**3.4 – 2 variable linear inequalities**

graph these inequalities

dashed or solid line?

shade up or down? left or right?

graph compound inequalities

same thing, but what part gets shaded by BOTH inequalities?

**3.5 – relations and functions**

different forms of relations

mapping diagram

set of ordered pairs

equation

graph

find domain and range of relations (from looking at any of the above forms)

from a graph: look left to right for domain; look bottom to top for range

definition of a function

decide if a relation is a function

if you see a graph -> use vertical line test

given a function, find the value of the function at a particular input

graph linear functions (just like linear equations, remember y = f(x)

**5.5 Divide Polynomials
**Divide by one term (monomial): divide each term on top one at a time (try to
cancel factors)

Divide by more than one term (polynomial):

Like long division

Remember what to do if you’re “missing” a term

**6.1 – 6.3 Factoring Expressions**

Take out a common factor from each term

Trinomials

Difference of squares (NOT sum):

Difference of cubes:

Sum of cubes :

**6.5 Solving Equations by Factoring (shows up at the end
of other problems too)
**As soon as you seex

^{2}and x :

Move every term to one side of the equation

Factor (using techniques from 6.1 – 6.3, depending on the problem)

Set each factor = 0, then solve each equation

**7.1 Rational (fractional) Expressions**

Find domain (remember denominator ≠0; set notation)

Write in lowest terms (factor, then try to cancel)

Multiply (straight across)

Divide (leave the first fraction alone, but multiply by the reciprocal of the second fraction)

**7.2 Adding and Subtracting Rational Expressions (just like adding/subt. any fractions)**

Need common denominator

Remember how to find LCD, then rewrite each fraction using that LCD

Add or subtract the numerators, keep the same denominator

**7.3 Complex Fractions (fractions inside of fractions)**

Remember what negative exponents mean (example: )

Simplify using one of two methods (I think the second is easier)

Method 1: simplify the numerator, simplify the denom.,

then multiply the numerator by the reciprocal of the denominator

Method 2: multiply by (you want to get rid of all of the little fractions), then simplify

**7.4 Solving Rational Equations**

1. Check the domain of x to know what solutions you have to eliminate

2. multiply both sides of equation by LCD (you want to get rid of fractions)

3. simplify, and solve for x

4. if any one of your solutions is not in the domain of x (check 1

^{st}step), then throw it out

**7.5 Applications**

Take a formula, and solve for one of the variables

Use proportions

Problems with distance, rate, and time

**5.1 Integer Exponents
**Product Rule

Quotient Rule

Power Rules (3 different ones)

Negative exponents

a

^{0}=1(for any a)

Scientific Notation

**8.1 Radical Expressions**

Find roots

Graph radical functions (set up x-y table, plug in x’s, then use calculator to help find y’s)

may help to find a root using your calculator (or for different problems in 8.2)

**8.2 Rational Exponents**

Remember the rule: to START simplifying expressions

Use it to write exponentials as radicals

Use it also to write radicals as exponentials

**8.3 Simplifying Radicals**

Product Rule

Quotient Rule

Rules for simplifying radicals (IMPORTANT):

1. no fractions in radicand

2. no radicals inside a denominator

3. the index and any exponents in the radicand must have no common factor greater than 1

4. the radicand must have no factor raised to a power ≥ the index

**8.4 Add/subtract radicals**

Simplify each radical FIRST, then treat any common radicals like a variable when you add/subt.

When fractions are involved, try to simplify each fraction FIRST, then add/subt. using the LCD

**8.5 Multiply/”Divide” Radicals
**FOIL binomials

To “divide” - rationalize the denominator

When the denominator is a binomial, multiply the numerator and denom. by the conjugate.

**8.6 Solving EQUATIONS with Radicals**

Step 1:Isolate one radical

Step 1:

**Step 2:**Use the power rule

**Step 3:**Solve the new equation

**Step 4**: Check EVERY possible solution in ORIGINAL equation

**8.7 Complex Numbers**

Use i to multiply, divide, and simplify radical expressions

Add/subtract complex numbers (combine like terms)

Multiply complex numbers (FOIL)

“Divide” complex numbers (use complex conjugate to get i out of the denominator)

**9.1 and 9.2 Solving Quadratic Equations (if you can’t/don’t want to factor)**

Sq. root property: As soon as you see x

^{2}an term or (but NO plain terms):

Isolate the squared part

Take the square root of both sides (remember ±)

If you’re not done already, solve for x

As soon as you see an x

^{2 }term AND an x term:

Move every term to one side of the equation (need: )

If you can’t/don’t want to factor – complete the square OR use quadratic formula

To complete the square:

1. If a≠1, divide every term by .

2. Move constant to the other side.

3. Take ½ of the number in front of x (remember you’ll use this answer later),

then square it.

4. Add that squared answer to both sides.

5. Factor the trinomial, but remember it should turn into

(x then that ½ answer from before.)

^{2}

6. Use sq. root property, and solve.

Memorize the quadratic formula:

Don’t forget, if you have a neg. INSIDE of a sq. root,
that means you take it out and put on the OUTSIDE of the sq. root.

**9.4 Applications of quadratic equations
**Solve formulas for a different variable (using the techniques above).

Solve word problems (using the techniques above; you may also need to use ).

**9.5 and 9.6 Graphs of Quadratic Equations (Parabolas)**

Remember how to graph (or really what it is/where it is on the coordinate plane).

Use that graph to shift up shift down shift left

shift right or a combination of shifts.

Look at the a in or …

a>0 parabola opens up a<0: parabola opens down

parabola is wider than

parabola is narrower than

For …

Vertex : (h, k) y-intercept: plug in 0 for x, and solve

For

Vertex: or y-intercept:(0, c)

Remember how to find domain and range from a graph

(domain of a parabola is always(−∞, ∞) , but you’ll have to think about range more).

**4.1 Systems of Linear Equations
**To check if a solution (x, y) is correct, plug into BOTH original equations

Solve any system by using substitution, elimination, or graphing

Know how to recognize when there is NO solution or when there is an infinite # of solutions

**4.3 Applications of Systems of Equations**

Step 1: read

Step 2: define variables for what you don’t know (usually what you’re asked to find)

Step 3: write equations

Step 4: solve with substitution or elimination

Step 5: check answers

Problems with cost:

Try to add up to a total # of items bought/sold

AND/OR

Try to add up to a total amount of $ made/spent

Problems with a mixture:

Try to add up to the total amount mixed/split up

AND/OR

Try to add up to the total concentration/interest/cost

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