Math Study Guide for the PERT

Algebra Concepts

Expressions, Equations, and Inequalities

Algebraic expressions consist of constants, variables, and exponents that are joined together by mathematical operations. Expressions can be monomials, binomials, or polynomials.

Expressions

Constants are numbers, such as $23$. It may sound silly to say it, but $23$ always means $23$, never $21$ or $15.7$. Because numbers have one and only one meaning, they are called constants.

Variables, on the other hand, are intended to act in place of numbers and can stand for any number at all. Once written into an equation, though, there are often only one or two numbers that will make the equation true. Much of algebra is determining the number(s) a variable must represent.

Exponents are usually constants that tell you how many of a certain number to multiply together. The number $4^3$, for example, indicates the multiplication of three fours, $4 \cdot 4 \cdot 4$, which is $64$. We can write $4^3=64$.

It’s also possible that a variable could be an exponent, such as $3^x$. You might see something like this: If $3^x = 9$, what is $x$? You can probably figure out that $x$ must be $2$, because $3^2 = 9$.

A combination of any of these values is called an algebraic expression. Certain fairly simple algebraic expressions are named based on their number of terms:

Expression	Name
$2x^2$	Monomial
$9x -3$	Binomial
$x^2+7x-5$	Trinomial

More broadly, any expression with two or more terms is a polynomial (binomials and trinomials are types of polynomials).

Operations With Expressions

When working with expressions, there are several operations you may be asked to perform on them. It is important to know the correct procedures for these.

Simplifying

You can simplify polynomials by combining like terms. The phrase “like terms” refers to terms that have variables and exponents that match. An example will make this clearer.

Simplify the polynomial:

\[3x^2 + 4x - 5x^2 + 16 - 7x\]

First, you have to rearrange the terms to group like terms together:

\[3x^2 - 5x^2+4x -7x +16\]

Now, by combining like terms, we get:

\[-2x^2 - 3x + 16\]

Adding and Subtracting

Adding polynomials can be quite simple. To do it, place like terms together and then add them as you would any other term. This can be done with two or more polynomials.

With subtraction, you have to be more careful because the signs of the terms in the subtrahend (the terms being subtracted) will all change:

\[(3a^2 + 4a + 25) - (2a^2 + 4a - 5)\] \[3a^2 + 4a + 25 - 2a^2 - 4a + 5\] \[3a^2 - 2a^2 +4a -4a +25 +5\] \[a^2 + 30\]

Multiplying

If you’re multiplying two binomials, you can use the FOIL method to keep track of the steps. FOIL stands for first, outside, inside, and last, which refer to the pairs of numbers being multiplied. For example, let’s multiply:

\[(x-5)(3x+8)\]

First: $x \cdot 3x = 3x^2$
Outside: $x \cdot 8 = 8x$
Inside: $-5 \cdot 3x = -15x$
Last $-5 \cdot 8 = -40$
Combine the terms and get $3x^2 -7x - 40$.

There is another slightly longer method that works too. Multiply each term of the first polynomial by each term of the second polynomial. Add all the resulting terms and simplify. Here’s an example:

\[(2x-3)(x^2 -5x +4)\] \[2x \cdot x^2 + 2x \cdot (-5x) +2x \cdot 4 -3 \cdot x^2 -3 \cdot (-5x) -3 \cdot 4\] \[2x^3 -10x^2 +8x -3x^2 +15x -12\] \[2x^3 -13x^2 +23x -12\]

To multiply quantities with exponents, simply add the exponents:

\[x^3 \cdot x^4 = x^7\]

In a more complex expression, it can help to rearrange it so that like terms are next to each other, then add the exponents:

\[\frac{1}{2} x^2y^3 \cdot 12 xy^2\] \[\frac{1}{2} \cdot 12 \cdot x^2 \cdot x \cdot y^3 \cdot y^2\] \[6 x^3y^5\]

Dividing

Dividing a polynomial by another polynomial can be simplified by first factoring each polynomial, canceling out similar terms or factors, and then simplifying.

Factoring

Factoring is the process of rewriting an expression in terms of its factors. Some binomials (expressions with two terms) can be factored if both terms share a common factor.

In the binomial $3x^2 +9x$, both terms have the factor $3x$, so the factored version will be $3x(x+3)$. This same trick can work for some trinomials or even higher-order polynomials.

Quadratic expressions can often be factored into the product of two binomials.

The quadratic $x^2 + x -20$ can be factored into the product $(x+5)(x-4)$. Some cubic equations can be factored into three binomials, but those are beyond the scope of this test.

Dividing by Monomials and Binomials

It is easiest to divide polynomials by a monomial. Do so by dividing each term of the polynomial by the divisor:

\[\frac {(12x^3 + 8x^2 - 2x + 6)}{4x}\] \[\frac{12x^3}{4x} + \frac{8x^2}{4x} -\frac{2x}{4x} +\frac{6}{4x}\] \[3x^2 + 2x - \frac{1}{2} + \frac{3}{2x}\]

Dividing by a binomial or another polynomial can require long division. This can sometimes be avoided by first factoring the polynomial (if possible):

\[\require{cancel}\frac {x^2 - 4x - 45}{x^2 + 6x +5} = \frac {(x-9) \cancel{(x+5)}}{(x+1) \cancel{(x+5)}} = \frac {x-9}{x+1}\]

Evaluating Expressions

Evaluating an expression involves substituting numbers for variables in the expression and doing the math necessary to simplify the expression, often to a single number. Consider the example below:

What is the value of $2ab + 5b^3$ when $a=3$ and $b=2$?

Substituting in the values of $a$ and $b$ gives us:

\[2 \cdot 3 \cdot 2 + 5 \cdot 2^3\] \[12 + 40\] \[52\]

Solving Equations and Inequalities in Algebra

There are many important guidelines for solving equations and inequalities in algebra. The procedures vary slightly depending on whether you deal with an equation or an inequality. There are additional strategies needed when dealing with special circumstances, such as simultaneous equations.

Linear Equations

Equations with variable exponents of $\bf{1}$ (or of the first degree) are known as linear equations. They are solved by manipulating the terms so that all the variable terms are on one side of the equation (often the left side), and the constant terms are on the other. This is done using the addition, subtraction, multiplication, and division properties of equality. Then, both sides are simplified to get a single variable on one side and a constant on the other. The basic rule is whatever you do to one side of an equation or inequality must also be done to the other. We’ll look at an example.

Solve $3x - 4 = x + 8$.

\[3x - 4 = x + 8\] \[3x - 4 +4 = x + 8 + 4\] \[3x=x+ 12\] \[3x - x = x - x + 12\] \[2x = 12\] \[2x \div 2 = 12 \div 2\] \[x = 6\]

Linear equations are the algebraic expressions of lines on a graph. A linear equation involves one or more variables, with the greatest exponent being $1$. Test questions may ask where two lines meet, or they may ask for the $x$ and $y$ coordinates where the two equations are true.

Simultaneous Equations

A system of linear equations can be solved to determine the number of times two lines intersect and the point where this intersection occurs. This means there is one solution that is simultaneously true for both equations. A system of linear equations can be solved using substitution, elimination, or graphing.

Equations With Two Variables

The two variables in a linear equation can be solved as long as there are two distinct equations given. Let’s look at an example.

Find the coordinates of the point of intersection for the lines $2y - 5x = 12$ and $y = 3x + 2$.

We’ll start by setting like the like terms of each equation in columns:

\[\begin{array}{rrrr} &-5x&+2y&=&12\\ &3x&-y&=&-2\\ \end{array}\]

To get down to just one variable, $x$, we’ll multiply the second equation by $2$ and add the equations, effectively eliminating the $y$ variable:

\[\begin{array}{rrrr} &-5x&+2y&=&12\\ &6x&-2y&=&-4\\ \hline &x& &=&8\\ \end{array}\]

Now that we know $x=8$, we can insert that into the second equation and get:

\[y = 3(8)+2 = 26\]

The coordinates where the two lines meet are ($8, 26$). This can be verified by substituting the $x$ and $y$ values into either equation and confirming the equality.

Linear Inequalities

Linear inequalities use the symbols $\lt$ (less than), $\gt$ (greater than), $\le$ (less than or equal to), and $\ge$ (greater than or equal to) instead of the equal sign. Manipulating the inequality is similar to equations except for the following:

Multiplication and division involving a negative sign change the direction of the sign of the inequality.
Swapping the values on the right and left of the inequality has the same effect.

Solve for the value of $a$:

\[2(3 - a) \le 8\] \[6 - 2a \le 8\] \[-2a \le (8 - 6)\] \[(-\frac{1}{2} \cdot (-2a) \ge 2 \cdot (-\frac{1}{2})\] \[a \ge -1\]

Another skill you may need is graphing inequalities on a number line. Let’s take a look at one such graph and figure out the inequality that it was based on:

2 Number Line for Inequality (fixed).png

The solid dot at the $-4$ position means that $-4$ is one of the values in the solution set. The solid line and right arrow at $2$ mean that all numbers to the right of the $-4$ are also included. (The arrow could have been shown anywhere to the right of $-4$$ and it would have meant the same thing.)

So, according to the number line above, $-4$ is included in the values, along with all numbers greater than $-4$. This inequality is written as:

If the values included everything greater than $-4$ but did not include $-4$, you would see a hollow dot at $-4$.

3 Number Line for Inequality 2 (added).png

\[x\geq-4\]

Quadratic Equations

Quadratic equations are equations with an exponent of $2$ (also known as second-degree equations), or equations that have their variables squared, hence, the term quad. The standard form of a quadratic is:

\[ax^2 + bx + c = 0\]

Factoring a quadratic is one way of solving for the values of the variable that make the equation true. Those values are called the roots of the equation. Let’s look at an example.

Find the roots of $x^2 - 9x + 20$.

First, we factor:

\[(x - 4)(x - 5) = 0\]

Then we set each factor equal to $0$:

\[x - 4 = 0 \,\text{and}\, x - 5 = 0\]

Solving for $x$ gives us the roots:

\[x=4 \,\text{and}\,x=5\]

Another way to solve a quadratic equation is to use the quadratic formula:

\[x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}\]

Solve $x^2 -x - 12 =0$ using the quadratic equation.

Comparing this equation to the standard form of a quadratic, $ax^2 + bx + c = 0$, we can see the values of $a$, $b$, and $c$:

\[a=1,\,\, b = -1, \,\,c = -12\]

Now, we’ll insert those values into the quadratic equation:

\[x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}\] \[x = \frac {1 \pm \sqrt {(-1)^2 - 4\cdot 1 \cdot -12}}{2\cdot 1}\] \[x = \frac {1 \pm \sqrt {1 +48}} {2}\] \[x = \frac {1 \pm \sqrt {49}} {2}\] \[x = \frac{1-7}{2} = \frac{ -6} {2} = -3\]

\[x = \frac{1+7}{2} = \frac{ 8} {2} = 4\]

So, $-3$ and $4$ are the roots of this quadratic equation.

Equations with No Constants

Equations don’t have to have constants to be solvable. An equation with only variables can be solved for any variable you choose. The formula for the area of a rectangle, for instance, is:

\[A = bh\]

where $A$ is the area, $b$ is the base, and $h$ is the height.

With this formula, we can solve for any of the individual variables by dividing or multiplying as needed. For example, to solve the given formula for the variable $h$, divide both sides by $b$ and the formula becomes:

\[h = \frac {A}{b}\]