PCAT Exam: Quantitative Reasoning - A Simple Guide To Algebra

Expressions, Equations, and Inequalities
In algebra, letters are used to symbolize quantities. If these quantities are known and fixed so that they cannot change, then they are called constants. If the quantity is not known, or if it is any value from of a set of arbitrary values from some domain, then it is called a variable. Any letters can be chosen to represent a constant or a variable. Usually letters from the beginning of the alphabet, like a, b, and c, are used to represent constants, and letters from the end of the alphabet are used to represent variables, such as x, y, and z.

An expression is any mathematical statement involving constants, variables, numbers, and operations between them.

Therefore, is an expression, and so is .

Expressions are the basic building blocks of statements that can be made in mathematics.
An equation is a mathematical statement that states 2 expressions are equal to one another.

For example: .
Given an equation, both sides of the equation can be multiplied or divided (by a nonzero quantity) by the same constant and get another true equation.

Both sides of the equation can also be added or subtracted by the same quantity to get another true equation.
An inequality is a mathematical statement in which one expression is stated to be greater than, greater than or equal to, less than, or less than or equal to a second expression.

In a strict inequality, such as , the two sides cannot equal one another.

In non-strict inequalities, such as , the two sides can equal one another.

When working with inequalities, it is possible to add or subtract the same quantity to or from both sides and get another true inequality.

However, when multiplying or dividing, some care must be used. When multiplying or dividing both sides by a positive value, the result is another true inequality.

However, when multiplying or dividing by a negative quantity, the direction of the inequality must be reversed in order to get another true inequality.

For example, given that , when multiplying both sides by -1, the inequality must be reversed to the other direction, to get .

Solving an equation is a common problem in algebra.

A problem gives some equation(s) involving one or more variables and requires finding the variable(s) that will make the equation(s) true.

To solve the equation, one generally tries to simplify the equation(s) involved until they are in a form in which one can read off the possible values for the variable.

It is also often useful to use the following rule: if , then either or , or possibly both.

Similarly, to solve an inequality, one must find all possible values of the variable(s) that make the inequality true.

The process of solving inequalities is very similar to solving equations. Specific techniques for solving equations are discussed below.

Evaluating Algebraic Expressions
Given an expression involving constants and variables, it is important to be able to evaluate these expressions for given values for the constants and variables.

To do this, substitute the constants and variables with the numerical values given for them.

This operation is perhaps most easily understood by looking at a few examples.
Consider the expression .

Suppose a problem asks for this expression to be evaluated when .

To do so, replace each instance of x with the value 2.

The expression becomes .

The next step is to apply the order of operations.

First, simplify the exponent: .

Then, apply the multiplication .

The last step is to subtract, which results in 10.

When one is substituting values into the expressions, and one is working with negative values, it is important to ensure that the entire value is substituted in for the variable or constant.

Consider the example of evaluating , but this time, for the case when .

 

To do this properly, one must ensure that the entire given quantity gets squared and subtracted. The result is .

This result simplifies to .
It is, of course, possible to evaluate expressions with multiple variables and constants.

For example, when .

As in the previous examples, replace each constant and variable with the given numerical quantity, which results in . The next step is to apply the order of operations. First, the multiplication is done, with the result of  .

The last step is to add, which results in 4.

Sometimes, a problem may require one to simplify an expression that has several constants and variables, and to give values only for certain constants or variables.

Consider the expression .

Suppose it is given that .

Since there is no given quantity for , is simply left as a variable, while all the instances of in this expression are replaced with 2.

The result of that replacement is .

As discussed below, there are many instances where this kind of simplification is utilized in solving algebraic equations and inequalities.

Representing Verbal Quantitative Situations as Algebraic Expressions or Equations
When dealing with a word problem (that is, a “verbal quantitative situation”), the first step is to translate the problem into an appropriate mathematical expression, equation, or inequality. This requires the use of both mathematical knowledge and critical reasoning about the problem itself. The general process is to begin by labeling each quantity involved with a constant or a variable, and then write down the relationships between them.

 

For example, suppose a problem deals with a situation in which shirts at a store cost $10 each, plus 5% sales tax, and the problem asks for an expression for the amount, in dollars, that a person will spend on an arbitrary number of shirts.

In other words, the number of shirts purchased is a variable that can be labeled as .

Now for each shirt, the person spends $10, plus 5% of $10 in sales tax. 5% of $10 is dollars.

The expression is the total money spent on shirts, which is , which simplifies to .

Now, suppose there is a problem in which a person is buying fruit; apples cost $1 each, and oranges cost $2 each.

Suppose the problem asks for an expression for the total money spent on fruit, assuming the buyer buys only apples and oranges.

Begin solving this problem by giving a label to each quantity.

Let represent the number of apples purchased and represent the number of oranges purchased.

For each apple, the buyer spends $1, and for each orange they spend $2.

Therefore, the total amount spent, in dollars, is .

 

Using Linear Equations and Inequalities
A linear equation with one variable is an equation with a single variable that can be simplified to the form .

Thus, is also a linear equation, since one could subtract the from both sides to get zero on the right side so long as and are held constant.

 

A linear inequality is just like a linear equation, except that it involves an inequality sign instead of an equals sign.

As previously mentioned, solving an equation means finding the values of the variable that make the equation true. The general approach is to try to isolate the variable on one side. For linear equations, this means to first move all the constants to one side and the variables to the other side.

Then, divide both sides by the coefficient of the variable.

 

Thus, consider the equation

.

Begin solving this equation by subtracting 1 from each side, which results in .

The next step is to add 4 to each side, which gives the equation .

Now, divide both sides by 6, which yields .
A linear equation with one variable always has a single solution, provided that the coefficient of is not zero.

Solving linear inequalities proceeds in almost exactly the same fashion.

However, it is important to keep in mind that multiplying or dividing by a negative number reverses the inequality.

 

Consider the example

.

To solve this inequality, start by subtracting from both sides.

This results in the inequality .

The next step is to divide both sides by -3.

However, this division reverses the inequality, which results in the new inequality

.

Absolute Value in Equations and Inequalities
The absolute value of a number is the positive distance of a number from zero.

If the number is positive, it is simply that number itself; if the number is negative, its absolute value is that number multiplied by negative one (-1).

Therefore, the absolute value is always a non-negative quantity.

The absolute value of is written as .

When dealing with equations that have an absolute value, one must split the solution into two cases: the case in which the quantity involved is non-negative, and the case in which it is negative.

 

Consider the equation

.

This equation can be split into two equations, each with a condition on .

First, if , the equation becomes , or , which has the solution .

This number is greater than or equal to zero, so it is a valid solution.

The second case is if .

Then the equation becomes , or , which has the solution .

However, care is needed at this point, because of the extra conditions on that were made when splitting the original equation into two.

This second equation is only equivalent to the original equation when .

Therefore, is, in fact, not a solution to the original equation.

The only solution to the original equation is .

Sometimes, the absolute value might be taken of a more complicated expression, for example, consider the equation .

Again, this equation can be split into two cases: first, if .

This case is equivalent to saying .

In this case, the original equation becomes .

This simplifies to the equation , or .

Since , this is a valid solution to the original equation.

The second case is when , that is, when .

In this case, the equation becomes , or .

Continuing the process of solving this equation results in .

Since , this is a valid solution to the original equation.

Therefore, this equation has two real solutions: and .

An absolute value inequality has two different forms.

The first form is the inequality .

The value of is less than a, but is also greater than -a.

For example, the inequality can be rewritten as and .

The two inequalities can be combined into one compound inequality: .

The inequality can be solved by first subtracting, with a result of .

The last step is to divide all values by 3, which yields as the solution set.
The second form is the inequality

The value of x is greater than a, but is also less than -a.

For example, the inequality can be rewritten as or .

Both inequalities can be solved by subtracting 1, with a result of or .

The last step is to divide all values by 3, which yieldsor as the solution set.

 

Using Equations and Inequalities Involving Rational Expressions
A rational expression is an expression that has the form , where and are both polynomials.

To solve equations or inequalities involving rational expressions, the expression is typically rewritten to get rid of the denominator; as a result, the problem becomes an equation or inequality involving polynomials.

One can then apply the techniques mentioned above to complete the solution.
For example, consider the problem .

One can start by multiplying both sides of the equation by .

This results in the equation .

Now this equation can be solved like any other linear equation.

Subtracting from both sides and subtracting 2 from both sides gives the solution .

Similarly, when dealing with inequalities, start by multiplying to eliminate the denominator.

This multiplication introduces only one complication, namely, determining whether the direction of the inequality is changed.

The direction might be changed on only a portion of the domain.

 

Consider the inequality .

The first step is to multiply both sides by , but care is needed regarding when this multiplication changes the direction of the inequality.

When , the result is the inequality , but when , the direction of the inequality changes, and the result is .

Case 1: , .

Then, and .

Clearly, this cannot be true, so there are no solutions from this case.

Case 2: , . Then, and .

This case has the solutions , which are the only solutions since there were not any solutions from the first case.

Solving Quadratic Equations and Inequalities
Algebraic expressions are built by adding and subtracting together monomials.

A monomial is a variable raised to some whole number power, multiplied by a constant (called the coefficient): , where a is any constant and n is a whole number.

A constant by itself is also a monomial.
A polynomial is a sum of monomials. If the highest power of in the polynomial is 1, the polynomial is called linear. If the highest power of is 2, it is called quadratic.
If the equation has the form , then it can be solved by adding b to both sides and dividing both sides by a to get

Then, the square root of both sides can be calculated.

However, when taking the square root, it is necessary to remember that squaring a number gives the same result for both that number and its negative.

When taking the square root of both sides, it is necessary to use a symbol to account for this fact: .

 

These are two separate solutions, unless b happens to be zero.
If a quadratic equation does not have a constant, that is, if the constant term is zero, it takes the form .

In this case, the can be factored out to get .

From this, the solution of this equation is and the solution to the linear equation is .

Given a quadratic expression that has been written in the form , one can attempt to factor the expression as where , and .

It is sometimes possible to guess such a pair of numbers by looking at the positive and negative factors for c.

In the expression , the factors of 6 are 1, 2, and 3.

By experimenting a bit, it may be noticed that , and .

This means the expression can be rewritten as .

To solve the equation , one could rewrite this equation as .

The solutions of this equation are the solutions to and , so the solutions are .

If, during this procedure, the two terms obtained after factoring are the same, then the equation has only a single root, called a double root, and the equation has only one solution.

For example, consider .

Applying the above procedure, this equation can be factored into .

This equation has only the one solution, , which is a double root.
In general, however, it may not be possible to easily guess a pair of numbers, A, B, that enable these kinds of expressions to be factored. In these cases, more general methods for solving quadratic equations are needed, methods that are sometimes a little longer than the ones given above.

The two methods that follow always work for any quadratic equation.

The first approach that can always solve a quadratic equation is called completing the square.

Suppose that a problem asks for the solutions to the equation

(if necessary, divide by the coefficient of the squared term first to get the equation into this form).

Then, some quantity can be added to both sides of the equation in order to make the left side have the form .

Specifically, would need to be added to both sides.

Here is an example to show how this works:

Suppose the equation is .

If the constant on the left side were 9, then this equation could be factored, since

Now, in order to get the constant on the left to be 9, 10 must be added to both sides.

This gives the equation .

It is now possible to factor the left side, which gives the equation

At this point, the square root can be taken on both sides (but remember, this introduces a ), with a result of .

Finally, subtract 3 from both sides to get two solutions: .

It is actually possible to perform the process of completing the square in a very general way to give a formula for the solutions to any quadratic equations. This formula is called the quadratic formula, and it can be used to solve any quadratic equation. However, because of its length, it can be easier to solve some quadratic equations by simply factoring them manually, rather than using the formula.

Here is a brief review of the derivation of the quadratic formula, since this can help in memorizing it.

The derivation starts with the most general form of a quadratic equation, .

First, divide both sides by a in order to begin the process of completing the square.

This division results in the new equation .

Next, subtract on both sides to get , and then add the quantity to both sides in order to complete the square.

This gives .

The left side is now the square of a single linear term, so it can now be factored.

Performing this factoring and simplifying the right side results in .

Next, take square roots on both sides to get .

At last, it is possible to solve for , which gives the quadratic formula:


It is not necessary to remember how to derive this formula, but the formula itself should be memorized.

Knowing where the formula comes from can help in remembering it, however.

Note that if , then the equation has only a single solution, and it is a double root.

To solve an inequality involving a quadratic term, one needs to use the fact that the value of a quadratic expression only changes sign when it passes through zero.

So, once it is known when the two sides of an inequality are equal to one another, it is possible to tell when one side is bigger than the other by checking in between each of the places where they are equal.
For example, consider solving the inequality .

Start by solving the equation .

This is equivalent to , or .

To determine when the inequality holds, then, one must only check the inequality for , , and .

By substituting -3 for , the inequality becomes , or , which is true.

So, this inequality is true for .

Similarly, substituting 3 for gives , or , which is true.

So, this inequality is true for .

However, checking the inequality by substituting zero for x gives the inequality , which is not true.

So, the inequality is false for .

It is also not true for , since this is a strict inequality.

Therefore, this inequality is true for and .

Radicals in Equations and Inequalities
When an equation or an inequality involves radicals, all the radicals must be moved to one side. Then, both sides can be raised to the appropriate power to get rid of the radicals.

Remember that the quantity inside a square root must be non-negative.

When dealing with inequalities, remember that multiplying both sides by a negative quantity reverses the direction of the inequality.
For example, consider .

The first step is to isolate the radical, so add 2 to both sides.

This addition results in .

Square both sides, and the result is , or .

When dealing with multiple radicals, proceed by first isolating one radical, squaring both sides to remove it, and then repeating this process to remove the remaining radicals.

 

Consider the equation

.

Start by subtracting 1 from both sides, isolating the radical on the left, which results in .

Then, square both sides:


or


Isolate the radical on the right: .

Next, square both sides, which results in .

This problem can now be solved by using the quadratic formula.
 

The process to solve is similar for inequalities.

For example, with , the first step is to isolate the radical, so add 2 to both sides.

This results in .

Then square both sides, which results in , or .
Another example is .

There are two inequalities.

The first inequality is the original problem.

The first step is to isolate the radical, so add 2 to both sides.

This addition results in .

Square both sides, which result in , or .

The second inequality is the radical, which has to be greater than or equal to zero since the radical must not be negative. In this case, the inequality is .

Square both sides; the result is or .

The solution set for the inequality is
 

Equations and Inequalities with Multiple Variables 
A system of equations is a collection of equations all of which must hold true at the same time.

Here are some basic rules to keep in mind when working with a system of equations.

- A single equation can be changed by doing the same operation to both sides, just as one could do if there were only one equation.
- Substitution: If one of the equations gives an expression for one of the variables in terms of other variables and constants, it is possible to substitute the expression into the other equations and replace the variable.

This means the other equations will have one less variable in them.
- Elimination: If there are 2 equations of the form , then it is possible to form a new equation , or .

One of the variables is eliminated from an equation.
In general, solving a system of equations involves using substitution and elimination to find the possible values for one variable, and then substituting those values into the original equations and using them to continue to solve for the possible values of the other variables.

The simplest system of equations is a linear system of two equations.

This means there are two equations in the form:




To solve linear systems of two equations, either substitution or elimination can be used. Both methods will be used to solve a system of equations.

Consider the following system of equations and solve by elimination first:


Start by multiplying the first equation on both sides by -2, which changes it into .

Next, add this equation to the second equation, which will eliminate the term: simplifies to .

This equation can be solved for , resulting in .

Now, substitute this value for into either of the original equations to solve for .

Substituting the result in the first equation results in .

This equation can be solved like any linear equation with one variable; first the equation becomes , then , and finally, .

So, this system of equations has the solution .
Start with the system of equations, and solve by substitution:




This time, solve the first equation to get an expression for in terms of .

To do so, subtract from both sides of the first equation, giving .

Next, divide both sides by -3, to get .

Next, substitute this value in the second equation for ; the second equation then becomes .

Since this equation only involves the variable , it is possible to solve it to find the -value, and then substitute this value for back into to find .

The approach used to solve the equation does not affect the final answer. Some students are more comfortable using substitution and some students prefer to use elimination. It is best to be familiar with both approaches, because sometimes there is an obvious elimination or an easy substitution to be made.

Note that if one of the equations in the system of two equations can be made to look identical to another equation in the system, then it is redundant.

The set of solutions is then all pairs that satisfy the other equation.

For instance, in the system of equations , it is possible to make the second equation into the first equation by dividing both sides by -4.

This means the solution set is all pairs satisfying , or .

To see whether one equation in a pair of linear equations is redundant, the simplest way is to rewrite each equation is in the form .

The equations are redundant if one is a constant multiple of the other when written in this form, and the two together are called a dependent system. If the equations are not redundant, they are called independent.

It is also possible for two equations to be inconsistent.

This happens when the system can be made into the form , with c and d being different numbers.

Another way to determine that the system is inconsistent is if, while trying to solve the system, one ends up with a contradictory equation, such as .

In general, for a pair of linear equations with two variables, there is always a single solution if they are consistent but independent. There is a complete line of solutions if they are redundant. Finally, there are no solutions if they are inconsistent.

Solving a linear system of three equations with three variables can be very similar to the solution of the case with two equations and two variables. Once again, the goal is to eliminate variables, using the method of adding equations to eliminate variables, or substituting expressions for one variable into another equation. However, this time there are three equations and three variables to keep track of.

Consider the following system:






Adding the first two equations will eliminate the term, obtaining the new equation .

Multiply the second equation by 2; the result is .

Add this result to the third equation, and the term is eliminated there as well, to get .

Subtract from this equation to get , or .

Now, substitute this value for x in .

This equation becomes , or .

Substitute both these values in the first, original equation to obtain , or .

The solution to this system is .

Functions
A function is defined as a relationship between inputs and outputs where there is only one output value for a given input.

Take for example an equation that says that . is a function of , since, for each value of , there is a unique value of y that satisfies the equation.

One useful way to express this is to use function notation, in which the function itself is given a label.

The letters that are used to label functions can, of course, be anything, but conventionally, functions are labeled with the letters f, g, and h.

To emphasize the fact that the function f takes the variable as its input, one can write , which is read “f of .”

If there is a formula for how to compute the value of from , this can be expressed by setting equal to this formula.

In the case of the above example, this would be the equation .
As another example, the following function is in function notation: .

The represents the output value for an input of . If , the equation becomes:






The input of yields an output of , forming the ordered pair .

The following set of ordered pairs corresponds to the given function: .
The set of values that is allowed to take in is called the domain of the function, and the set of possible outputs is called the range of the function.

By definition, each member of the domain is paired with only one member of the range.

Unless a problem specifies otherwise, it is usually assumed the domain is all real numbers other than those for which the expression for is not defined.

For example, consider the function , the value cannot be part of the domain (because the denominator would be zero).

Unless the problem specifies a more restricted domain, the domain of this function is all real numbers except .

Functions can be represented multiple ways:



Interval Notation
The domain of a function, as well as the range, is often expressed by using interval notation.

 

Consider the function

.

This function is defined only when , or .

This function can be expressed in interval notation as .

 

Here are some other examples of interval notation.

can be written as .

can be written as .

or can be written as .

Square brackets indicate non-strict inequalities, and curved parentheses indicate strict inequalities.
If a variable is a function of , then is called the independent variable and is the dependent variable.

These names come from the fact that in such a problem one often begins with some value of and then determines how depends upon that starting value.
To denote the value a function takes for some specific value of , write that value in place of in the function notation.

Thus, indicates the value that this function will take when is equal to 3.

If , then .
Evaluating functions is exactly like evaluating expressions: substitute the given value into the expression for the function, and perform the indicated arithmetic operations using the order of operations.

Performing Algebraic Operations on Functions
Given a pair of functions and , it is possible to form new functions by performing arithmetic operations between the functions.

For example, one can get the new function .

If the functions are and , then it is possible to compute an expression for by adding together those functions. For example, suppose that and .

Then the expression for .
Similarly, one can subtract functions: .

When finding an expression for such subtraction of functions, it is very important to make sure to subtract the entire second function from the first function.

With the functions from the previous paragraph, the subtraction is:

Multiplying functions is accomplished similarly: .

Once again, when working out the formula for such a product, make sure to multiply both expressions completely.

Using the same f and g, the result is .
Finally, it is possible to divide functions. By definition, .

With the above examples, .

 

Determining Compositions of Functions
There is another way in which two functions can be combined, which is not an arithmetic operation.

This is by composing two functions. Given two functions f and g, the composition of g with f is written , and it is defined by the equation .

In other words, first apply the function f to , and then apply g to the result.

The domain of such a function is some subset of the domain of f.

Specifically, it is those values in the domain of f, such that the result when f yields something in the domain of g.
As a first example, consider the case where and .

Suppose a problem asks for .

To do this, one must substitute for in the formula for g. This substitution results in .

Now, this can be plugged in the formula for f, which gives .

This can now be simplified to give .

In this case, because both functions have a domain consisting of all real numbers, the composition has a domain of all real numbers as well.

Here is an example where care is needed regarding the domains.

Consider the case where .

Computing by using the same approach first gives , and then .

The domain of f consisted of all real numbers, but this new function is defined only when , that is, when , or .
 

Determining Inverses of Functions
A function f is called “one to one” if, in the equation , there is a unique value of for each unique value of .

In other words, f is one to one if, whenever , it is the case that .
If a function f is one to one, then it is possible to define an inverse function for f.

This is a function written as , which is defined so that .

One way to find an inverse function is the following.

Given a function f, write out the equation , and then solve for .

If f is one to one, the result is an equation of the form .

The expression on the right side of this equation yields the formula for the inverse of f.

Here is an example of this procedure.

Suppose that .

To find the inverse of f, apply the method given in the previous paragraph.

Start by writing out the equation .

Now solve for which gives , or . So .
Functions that are not one to one for all real numbers may be one to one on a restricted portion of their domain.

For example, is not one to one, because .

However, if the domain is restricted to non-negative , then f is one to one, and it is possible to find an inverse: gives us .

This gives two possible values, which does not define a function.

However, one may now use the fact that the original f only took values of that were non-negative.

This means that one does not need to consider the possibility. Therefore, .

Determining Maximum and Minimum Points
Finding the maximum and minimum points of general functions requires the use of calculus.

However, for functions such as quadratics, it is possible to write down a general formula for the maximum or minimum point of the function.
Every quadratic function can be written in the form .

The graph of a quadratic function is defined as a parabola.

Parabolas are vaguely U-shaped. If a is positive, then the U-shape opens upwards; if a is negative, the U-shape opens downwards.

Parabolas that open upwards have a minimum point; parabolas that open downwards have a maximum point.


This high or low point is called the vertex.

Note that a parabola is always symmetric about a vertical line (axis of symmetry). In the graphic above, the axis of symmetry is the line .

To graph a parabola, its vertex and at least two points on each side of the axis of symmetry need to be determined.
Given a quadratic function in standard form, , the axis of symmetry for its graph is the line .

For the quadratic function , and .

The axis of symmetry can be calculated as follows: , so is the axis of symmetry.

The vertex for the parabola has an x-coordinate of .

To find the y-coordinate for the vertex of the previous example, the calculated x-coordinate needs to be substituted into the original function.

Because the axis of symmetry passes through the vertex, the x-coordinate is -2.
The y-coordinate is .

The vertex is .

 

To complete the graph, two different x-values need to be selected and substituted into the quadratic function to obtain the corresponding y-values. This will give two points on the parabola. These two points and the axis of symmetry are used to determine the two points corresponding to these. The corresponding points are the same distance from the axis of symmetry (on the other side) and contain the same y-coordinate. Plotting the vertex and four other points on the parabola allows for constructing the curve.
 

When finding minimum or maximum values, the y or f(x) value is the only value needed for the answer. Whether that value represents the minimum or maximum, depends on if the parabola opens upwards or downwards.

For example, the height of a baseball is given, in feet, by .

Suppose the problem asks for the highest height the baseball will reach.

Clearly it does have some maximum point because the coefficient of is negative.

The time value (t) in this example represents the x-coordinate of the vertex.

This value needs to be found first using the formula where .

The value of t is .
Substituting this value into the expression gives the maximum height of the baseball: