Download A quick overview of expressions

NOTES: A quick overview of algebraic expressions
algebraic expression:
 a mathematical phrase that can include variables, constants, and operations
 does NOT include an equals sign (=)
variable (letter that stands for a number)
constant (number without a variable)
2y + 5
operation
coefficient (number directly in front of a variable)
• means that number times the variable
• can be positive or negative; can be an integer, decimal, or fraction
• If there’s no coefficient, it’s an invisible 1: x = 1x –x = –1x
How do you translate from words into algebraic expressions?
Addition
words
five more than a number
expressions
n+5
ten minus a number
10  n
ten less than a number
n  10
four less than three times a number
twice a number
3n  4
2n
half of a number
1
plus, added to, increased by,
sum, total, all together, more than
Subtraction
minus, subtracted from,
decreased by, difference
less than, fewer than
Multiplication
times, multiplied by, product, half
of, doubled, twice, tripled
Division
divided by, divided into, quotient,
per, ratio of
Exponents
𝑛
𝑛 (=2)
2
the quotient of 6 divided by a number 6
𝑛
𝑛
the quotient of a number divided by 4
4
a number squared
n2
squared, times itself, cubed
Parentheses
the quantity of,
times the sum/difference of
four times the quantity n minus five
4(n  5)
twice the difference of x and y
2(x  y)
D. Stark 1/102017
1
How do you combine like terms?
 A term is a number or a variable or the product of the two. 3, x, and 3x are all
terms. Terms are added or subtracted in algebraic expressions.(*)
 You can group and count only things of the same sort:
cows with cows, sheep with sheep, goats with goats…
o When adding and subtracting algebraic expressions, we need like terms.
We can add 2 xylophones (2x) and 3 xylophones (3x) to get a total of 5
xylophones (5x). But we can’t add that to 4 yaks (4y) and get 9 of
anything.
 Like terms have the same variables raised to the same exponents.
Constants (numbers without variables) are also like terms for each other.
3 and 5 are like terms
2x and –6x are like terms
3x2 and 4x2 are like terms
2xy and 6xy are like terms
3 and 5x are not like terms
2x and –6y are not like terms
3x2 and 4x are not like terms
2x2y and 6xy2 are not like terms
 To combine like terms you add or subtract the coefficients (and the constants).
o If there’s no coefficient listed, the coefficient is 1; it’s there but invisible.
o It’s useful to collect all of one kind of term first.
When you do so, take all + or – symbols in front of the variable with you.
EXAMPLE: 2x  6y + 4 + 5x  3y
2x
 6y
+4
+ 5x  3y
= 2x + 5x
 6y  3y
= 7x
 9y
= 7x  9y + 4
+4
+4
EXAMPLE: 6x2 + 4x  5y + 7 + x  7x2  10
6x2 + 4x  5y + 7 + x  7x2  10
= 6x2  7x2
+ 4x + x
 5y
+ 7  10
2
= x
+ 5x
 5y
+ (3)
2
= x + 5x  5y  3
[Simplify “plus a negative” to subtraction at the end.]
(*) Some books insist that terms are only added in expressions, while others say they can be added or subtracted.
Really the difference is between writing subtraction (for example, 5x  4y  3 ) and writing addition of a negative
(5x + 4y + 3). It amounts to the same idea. Use whichever formulation works best for you.
D. Stark 1/102017
2
How do you use the distributive property?
[REALLY IMPORTANT!]
 The distributive property lets you simplify expressions like these:
2(3x + 4)
2(3x – 4)
(3x + 4)2
(3x – 4)2
o It applies whenever you have an addition or subtraction inside
parentheses and a number outside to multiply the whole thing by.
 Usually the outside number is on the left.
o It works with all integers, decimals, and fractions.
o If the parentheses include a sum or difference without variables,
you can do what’s inside first or you can use the distributive property. Do
whichever is easier. For mental math, the distributive property may be
best. With a calculator, doing what’s inside is likely quicker.
LEVEL 1 distributive property examples

The idea is to distribute the number outside to each bit of what’s inside.
That way each piece inside the parentheses gets multiplied.
2(3x + 4)
= 2(3x) + 2(4)
= 6x + 8
2
3x
4
6x
8
2(3x – 4)
3x
4
= 2(3x) – 2(4) 2
= 6x – 8
6x
8
 Keep the original addition or subtraction sign as your answer’s operation.
Do any simplifying of negatives at the end:
change any “– –“ to “+” and any “plus a negative” to subtraction.
–4(5x + 3)
–4(5x – 3)
= –4(5x) + –4(3)
= –20x + (–12)
= –20x –12
= –4(5x) – –4(3)
= –20x – (–12)
= –20x + 12
LEVEL 2 distributive property examples
 You can use the distributive property with multiplied variables, too.
You’ll need to use the rules of exponents.
–2x(4x2 + y)
= –2x(4x2) + –2x(y)
= –8x3 + –2xy
= –8x3 – 2xy
–2y3(5x – y2)
= –2y3(5x) – –2y3(y2)
= –10xy3 – (–2y5)
= –10xy3 + 2y5
= 2y5 + –10xy3
= 2y5 – 10xy3
D. Stark 1/102017
3
What does it mean to simplify an expression?
 Use the distributive property if applicable.
 Combine all like terms.
 Put the expression in standard form:
o Put higher order exponents on the left, then regular variables, then
constants. For example: 2x3 + 4x2 + x + 5 instead of x + 5 + 2x3 + 4x2
o Put in alphabetical order any variables that are stuck together.
For example: write 2xy instead of 2yx
o Change any “– –“ to “+” and any “plus a negative” to subtraction.
o Drop any coefficient of 1: 1x  x.
How do you evaluate an expression?
“Plug in & crank out”
(To evaluate an expression means to find its value, given the values of the variables.)
 First plug in all the values on one line to replace the variables.
o Use parentheses around the values you substitute.
This will indicate multiplication and mark negative numbers.
 Then crank out the rest with the order of operations, doing one step per line.
EXAMPLES:
Evaluate 6x – 3y + xy
for x = 4 and y = 2
Evaluate 3x2 – 2y2 + xy for x = 5 and y = –3
6x – 3y + xy
= 6(4) – 3(2) + (4)(2)
= 24 – 6 + 8
= 26
How do you apply a formula?
3x2 – 2y2 + xy
= 3(5)2 – 2(–3)2 + (5)(–3)
= 3(25) – 2(9) + (5)(–3)
= 75 – 18 + (–15)
= 57 + (–15) = 42
“Plug in & crank out”
 A formula has an equals sign (=) so it’s an equation, not an expression.
Still, you apply formulas just the way you evaluate expressions:
plug in the given values for variables, and crank out the order of operations.
EXAMPLE: If the temperature in Fahrenheit is 78, to the nearest whole degree,
5
what is it in Celcius? Here’s the conversion formula: C = (F – 32)
9
5
C = (F – 32)
Don’t be scared off by word problems that
9
present an unfamiliar formula!
5
C = (78 – 32)
9
Much of the time all you need to do is
5
C = (46)  26C
“plug in and crank out.”
9
D. Stark 1/102017
4
When are expressions undefined?
There are 2 undefined cases you might encounter on the GED test:
1) Within the world of real numbers, √𝒙 is undefined
when x is a negative number
FOR THE CURIOUS… What’s the problem with calculating √−9 ?
Well, remember what it means to find a square root.
Here you’re looking for a number (the SAME number)
that you could put into the blanks to make a true statement.
_____  _____ = –9
Now +3 won’t work since 3  3 = 9.
But –3 won’t work either since –3  –3 = 9, not –9.
That’s the problem.
What’s up with the phrase, “within the world of real numbers”?
Are there unreal numbers?
Actually, yes. There are imaginary numbers. The imaginary number i is
defined as √−1. And there are lots of real world uses for imaginary
numbers. But all that is beyond the scope of the GED test. As far as the
test is concerned, you can’t take the square root of a negative number.
D. Stark 1/102017
5
2) division by 0 (or having 0 in the denominator)
What values of x would make the expressions below undefined?
In other words, what values of x would make the denominators 0?
(These values of x are sometimes called excluded values since they’re illegal.)
LEVEL 1 examples
5
𝑥
x can’t be 0.
9
𝑥2
x can’t be 2 (because x  2 would = 0 if x were 2)
LEVEL 2 examples
7
(𝑥  2)(𝑥 + 3)
x can’t be 2 or 3
(because x  2 would = 0 if x were 2 and x + 3 would = 0 if x were 3
and the whole denominator = 0 if either of the parentheses = 0 since 0 times anything = 0)
10
𝑥(𝑥  5)
x can’t be 0 or 5
FOR THE CURIOUS… Why is dividing by 0 mathematically illegal?
Notice that you can move from any multiplication problem to division ones.
For example, from
2  3 = 6 we can get 6  2 = 3 and 6  3 = 2
What about 2  0 = 0? We’d get 0  2 = 0 and 0  0 = 2
The 1st is OK. If I have no money to split between 2 kids, each gets $0.
But the 2nd looks weird. Where’s the 2 coming from in 0  0 ?
What about 5  0 = 0? We’d get 0  5 = 0 and 0  0 = 5
Again, the 1st division is OK. If I split $0 among 5 kids, each still gets $0.
But the 2nd division now says 0  0 = 5. Which is it—2 or 5?
And I can keep going. I picked 2 and 5 at random. It turns out that by the
rules relating multiplication and division, 0  0 would come up at the same
time with infinitely many different solutions. And that’s not a happy situation.
So maybe it’s just a problem with 0  0. What about 8  0? Why couldn’t we
do that? Let’s say 8  0 = y. But then just as we can move from 6  2 = 3 to
saying 2  3 = 6, we’d have to be able to say 0  y = 8. But there’s no value
of y that would make that true. Dividing by 0 is just undefined.
GED® is a registered trademark of the American Council on Education (ACE) and administered exclusively by GED
Testing Service LLC under license. This material is not endorsed or approved by ACE or GED Testing Service.
D. Stark 1/102017
6