Welcome to the EQUATIONS (INEQUALITIES) - WORD PROBLEMS instruction unit.

 

These instructional quizzes are short, limited to only about a dozen questions, and they are not graded or timed. Occasionally there will be a question introducing a topic that you are not expected to answer at first. Do not panic. The instructional material will likely be introduced in the following answer explanation. The purpose of these instructional modules is to review subject material by topic that you will likely encounter on the exam and provide some topic specific tips to improve your ability to solve related problems. 

 

A MATHEMATICAL EXPRESSION CAN CONTAIN....?

 

 

Manipulating Expressions 

Commutative Property: 

Permits changing the order of terms in the expression without changing the result. It applies to...

• Addition & Multiplication are commutative e.g. 2 + 5 = 5 + 2 or 2 × 5 = 5 × 2
• NOT division, subtraction 2 - 5 does NOT EQUAL 5 - 2

Distributive Property: 

Permits "multiplying out" or re-distribution of multiplication across terms added or subtracted without changing the result. e.g. 3 × (4 + 2) = 3 × 4 + 3 × 2 

Associative Property: 

Permits changing the grouping of terms in the expression without changing the result. It applies to...

• Addition & Multiplication are associative e.g. 2 + (5 + 7) = (2 + 5) + 7 or 2 × (5 × 7) = (2 × 5) × 7
• NOT division, subtraction (2 - 5) - 7 does NOT EQUAL 2 - (5 - 7)

Adding & Subtracting Zero: 

Permits introducing new terms that will not change the value of the expression because the change is zero.

• e.g. 2 + y = 2 + y + b - b where (b - b) = 0
• may be useful with associative law if (y + b) is known and b is known, but y is not.

Multiplying & Dividing By One: 

Permits introducing new terms that will not change the value of the expression because 1 times "anything" is still "anything".

• e.g. 2 + y = (2 + y) × (b/b) where (b/b)=1
• may be useful with associative law if (y /b) is known and b is known, but y is not.

 

It is easy to discount the importance of these laws, but they are powerful techniques to transform an expression from something with little meaning into something that is known or extremely useful. 

 

What is 4(3 + y) - (5 - 3y)? 

 

A perfect square is the square of any integer.  

Is the product of two perfect squares still a perfect square in all cases? 

 

Leroy makes twice what Janice makes and Paola makes three times as much money as Janice. Paola makes $240. 

How much money does Leroy make? 

 

MANIPULATING EQUATIONS

 

The goal of performing addition or subtraction on both sides of the equation is to isolate and solve the equation for a particular variable. 

Step 1:  

How might you use addition or subtraction to simplify the following equation and isolate y? 

3y + 5 = 35 

 

Step 2:  

How might you use multiplication or division to simplify the following equation and isolate y?

 

3y = 30 

 

Almost any function or operation (not just add, subtract, multiply, and divide) can be performed in order to simplify the equation as long as it is done on both sides. What operation might best help evaluate the following equation? 



 

 

Cross multiplication is especially useful for ratio problems of the form a/b = c/d.  Multiply one denominator by the other numerator and vice versa as follows. 

a × d = c × b.  

If you think about it, cross multiplication is essentially combining two steps of multiplying both denominators on both sides into one step. 

Evaluate y in the following equation: 

3/4 = 6/y 

 

Combining like terms helps simplify equations and isolate one variable, often the variable to solve for, from other values and terms.  

Evaluate x in the following equation: 

3x + 5 = 2x - 3.