In the English language, we"ve acquired phrases and we"ve got sentences. A phrase is a wire of words the expresses a thought but doesn"t type a finish sentence. A sentence is a grammatically exactly string of native that contains both a subject and a verb. This is a sentence. Expression over here.

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Some Phrases

1. "the fat cat" 2. "purple feathers" 3. "upon the mountain" 4. "the bald eagle"

Some Sentences

1. "The fat cat preferred mice." 2. "The bird has purple feathers." 3. "The hermit lived upon the mountain." 4. "The outright eagle to be disappointed to find that the Rogaine wasn"t working."

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Not every cable of words is a expression or sentence. Part strings the words are nonsense. Favor a great majority the the poppycock the comes twaddling out of our claptraps.

Some Nonsense

1. "fish dog cat and also the" 2. "like swim eat chocolate" 3. "so if but or to"

We apologize if any kind of of this look suspiciously prefer the message messages friend send ~ above a everyday basis. We additionally apologize if the non-phrase "like swim eat chocolate" is ever the heading on your future digital dating profile.

In algebra, rather of phrases and also sentences, we have expressions and equations. One expression is a wire of mathematical signs representing a quantity. One equation is a wire of mathematical icons stating the equality of two expressions. Therefore, an expression is like a phrase and an equation is like a sentence. Instead of subjects and verbs, we have actually constants and variables. Instead of punctuation, we have symbols. We could go on choose this every day.

Some Expressions

1. x 2. 3y + 2x − 45(6 − x)2 3. 0

Some Equations

1. x = 2 2. 4x = 20 3. 3x + 10y − 12 = 0

Notice that expressions execute not have actually equal signs, when equations must have equal signs. One equation is a statement that 2 expressions have the same value, and also the equal sign is a necessity. "Equation" even starts through the exact same letters together "equal," which have to be a an excellent clue.

In order to be an expression or equation, the string of symbols should make sense. Catch you drift our?

Some Nonsense

1. xy − + − 34 2. + + + 3. x 2 ÷ ( ( (

The piece of one expression be separate by + and − indicators are called terms. A term will certainly be hopeful if it complies with a + authorize and negative if it follows a − sign. When we break an expression into terms, we should be careful to store the an adverse signs in former of state that room being subtracted. We additionally need come be careful not to gain our thumb in the means when we"re striking it v our ball-peen hammer.

Sample Problems

1. In the expression 3x + 2, the terms space 3x and also 2.

2. In the expression 3x − 2, the terms room 3xand -2. That"s because 3x − 2 way 3x + (-2). Don"t forget that there"s in reality a plus authorize in prior of optimistic numbers, even if we can"t always see them. We knew us never need to have given them that cloak that invisibility.

3. The terms in the expression are 5x, 3y2, and .

Expressions that are multiplied with each other are dubbed factors. You "ll need to concerned grips through this information—it"s just one the the factors of life.

Sample Problems

1. In the expression 3x, 3 and also x room both factors.

2. In the expression (3 − x)(2y2 + 9), (3 − x) and also (2y2 + 9) space both factors.

3. 5, x, and y are all factors of 5xy.

Rearranging Expressions


Two expressions are said to it is in equivalent if they produce the very same number for any possible value of the variable. We"ll gambling you desire to view some instances of this ingredient in action. Don"t ask exactly how we knew. We had a feeling.

Sample Problem

The expression x and x + 0 space equivalent, because for any kind of value of x, x and also x + 0 room the very same thing. Sorry, zero, however you"re nice worthless.

One means of to express the truth that these two expressions are tantamount is to create an equation: x = x + 0. Any time you have actually an equation, the two expressions on either next of the equal authorize are equivalent. That"s kinda the same sign"s totality deal. That doesn"t evaluate it when you write something prefer 1 + 1 = 3. Drives it batty.

Sample Problem

The expression x2 and also (-x)2 room equivalent, because for any kind of value that x we"ll gain the exact same value out of one of two people expression. If x is 2, we uncover that 22 = 4 native the very first expression and also (-2)2 = 4 native the 2nd expression. If x is 34,792, we get...oh, well, friend probably got it native the very first example. No should wear ourselves out.

The score is to develop a toolbox of ways to modify an expression come an tantamount expression. We"re not, that course, talking about a literal meaning toolbox. You"ll never ever receive an emergency contact to hop over and fix one overflowing toilet v your trusty ratcheting box variable. Instead, this moves room the rule we have for working through algebraic expressions. When we finish occurring this (virtual) toolbox, we"ll usage our tools to settle equations and more. Okay, possibly the sometimes leaky faucet, but that"s wherein we attract the line.

There are several various things we deserve to do come rearrange expressions. We"ve separated them into the adhering to list of allowable moves:

1. Commuting 2. Associating 3. Distributing 4. Factoring 5. Combining choose terms 6. Obtaining rid the parentheses

Use any type of moves that aren"t in the "toolbox," and also you can draw a penalty flag. Not correct Rearrangement that Expressions: 15 yards and loss of down.

Using combos the those moves with each other to rearrange an expression into an tantamount expression is calledsimplification.


This dominion for rearrangement is virtually so obvious it isn"t also worth mentioning. However we"re suckers because that overkill, so below goes. The commutative property says that the order in i m sorry we add or multiply numbers doesn"t matter.

x + y = y + x

xy = yx

This move, transforming the stimulate of our terms once we"re adding or multiplying, is one of the most simple ways we can rearrange an expression. Note that we don"t to speak you can also do this relocate with individually or division. That"s due to the fact that you can"t. Addition and multiplication space a little more go-with-the-flow; individually and department aren"t quite as easygoing and could more than likely stand come chillax a bit.


Sample Problem

Consider the expression -4y2x + x3. We can use the commutative building of addition to rewrite the totality thing as x3 – 4y2x.

We can also use the commutative residential property of multiplication to rewrite -4y2x as -4xy2.

Putting the together, we deserve to rearrange -4y2x + x3 utilizing commutativity (and yes, us did make that word up) to gain x3 – 4xy2. We"re left with an expression the doesn"t feature any type of fewer terms than the original, yet at least it doesn"t begin out through a an unfavorable sign. The really chaps our patootie.


Another preeminence for rearrangement shows up as soon as we add or multiply an ext than two terms. If you ever before have occasion to introduce these state to anyone in a service setting, you deserve to make yourself sound über-professional by referring to them together your "associates." for this reason the name: the associative property. Here"s what it looks prefer for addition:

(x + y) + z = x + (y + z)

And it additionally works because that multiplication:

(xy)z = x(yz)

This is one more one of this straightforward rules because that rearrangement that we use all the time. It"s a well line between obvious and also oblivious, and also we wanted to make sure.


Sample Problem

Say you"ve obtained nine females dancing, eight maids a-milking, 7 swans a-swimming, 6 geese a-laying, five gold rings, 4 calling birds, three French hens, 2 turtle doves, and also a partridge in a pear tree, and also you want to know just how many complete things the is. No to point out where in the civilization you"re gonna save them all.

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1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

Now we deserve to rearrange all those terms and group together the numbers room straightforward to add. Us rewrite utilizing commutativity and also associativity come get:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 =10 + 10 + 10 + 10 + 5 = 45

That"s 45 complete Christmas-y things. You have to be getting into the vacation spirit. Also if the roof of your auto is covered in French hen droppings.