Home > Articles

Math Section

This chapter is from the book

This chapter is from the book

Algebra

This book assumes a basic understanding of algebra. Our focus will be on reviewing some general concepts and applying those concepts to questions that might appear on the PSAT math sections.

Understanding Linear Equations with One Variable

In a linear equation with one variable, the variable cannot have an exponent or be in the denominator of a fraction. An example of a linear equation is 2x + 13 = 43. The PSAT will most likely require you to solve for x in that equation. Do this by isolating x on the left side of the equation, as follows:

  • 2x + 13 = 43

  • 2x = 43 – 13

  • 2x = 30

  • x = , or 15

Understanding Systems of Equations

A system of equations is a set of equations that is satisfied by the same set of values of the variables. On the PSAT, a system of equations will generally take the form of two linear equations with two variables. Following is an example of a system of equations, and the steps necessary to solve it:

  • 4x + 5y = 21

  • 5x + 10y = 30

The first step will be to make either the x-values or the y-values cancel each other out. In the above system, it will be easier to work with the y-values, since 5 is a factor of 10. Multiply each element in the first equation by -2:

  • -2(4x) + -2(5y) = -2(21)

  • -8x – 10y = -42

Now, you can add the like terms in each equation:

  • (-8x + 5x) = -3x

  • (-10y + 10y) = 0

  • -42 + 30 = -12

  • -3x = -12

Therefore, notice that the two y-terms cancel each other out. Solving for x, you get x = 4. Now, choose one of the original two equations, plug 4 in for x, and solve for y:

  • 4(4) + 5y = 21

  • 16 + 5y =21

  • 5y = 5

  • y = 1

The solutions for the system of equations are x = 4 and y = 1.

Understanding Polynomial Operations and Factoring Simple Quadratic Expressions

A polynomial is the sum or difference of expressions like 2x2 and 14x. The most common polynomial takes the form of a simple quadratic expression, such as: 2x2 + 14x + 8, with the terms in decreasing order. The standard form of a simple quadratic expression is ax2 + bx + c, where a, b, and c are whole numbers. When the terms include both a number and a variable, such as x, the number is called the coefficient. For example, in the expression 2x, 2 is the coefficient of x.

The PSAT will often require you to evaluate or solve a polynomial by substituting a given value for the variable.

For example: If x = -2, what is the value of 2x2 + 14x + 8 = ?

  • 2(-2)2 + 14(-2) + 8 =

  • 2(4) + (-28) + 8 =

  • 8 – 28 + 8 = -12

You will also be required to add, subtract, multiply, and divide polynomials. To add or subtract polynomials, simply combine like terms, as in the following examples:

  • (2x2 + 14x + 8) + (3x2 + 5x + 32) = 5x2 + 19x + 40

  • (8x2 + 11x + 23) – (7x2 + 3x + 13) = x2 + 8x + 10

To multiply polynomials, use the Distributive Property to multiply each term of one polynomial by each term of the other polynomial. Following are some examples:

  • (3x)(x2 + 4x – 2) =(3x3 + 12x2 – 6x)

Remember the FOIL Method to help solve some polynomial problems: multiply the First terms, then the Outside terms, then the Inside terms, then the Last terms.

For example: If (2x + 5)(x – 3) = ax2 + bx + c for all values of x, what is the value of b?

  • First terms: (2x)(x) = 2x2

  • Outside terms: (2x)(-3) = -6x

  • Inside terms: (5)(x) = 5x

  • Last terms: (5)(-3) = -15

Now put the terms in decreasing order:

  • 2x2 + (-6x) + 5x + (-15) =

  • 2xx – 15

Therefore, the value of b is 1, because the coefficient of x is 1.

You might also be asked to find the factors or solution sets of certain simple quadratic expressions. A factor or solution set takes the form, (x ± some number). Simple quadratic expressions will usually have 2 of these factors or solution sets. The PSAT might require you to calculate the values of the solution sets.

For example: If (2x + 5)(x – 3) = 0, what are all the possible values of x?

Set both elements of the equation equal to 0.

  • (2x + 5) = 0

  • (x – 3) = 0

Now solve for x:

  • 2x + 5 =

  • 2x = -5

  • x =

  • x – 3 =

  • x = 3

The possible values of x are and 3.

Following are some general factoring rules that might prove useful for answering PSAT math questions:

  • Finding the difference between two squares: a2b2 = (a + b)(ab)

  • Finding common factors, such as: x2 – 2x = x(x + 2)

  • Factoring quadratic equations, such as: x2 +2x –8 = (x + 4)(x – 2)

Understanding Linear Inequalities with One Variable

Linear inequalities with one variable are solved in almost the same manner as linear equations with one variable: by isolating the variable on one side of the inequality. Remember, though, that when multiplying both sides of an inequality by a negative number, the direction of the sign must be reversed.

For example: If 3x + 4 > 5x + 1, then x =

First, isolate x on one side of the inequality.

  • 3x – 5x > 1 – 4

  • -2x > -3

Now, since you have to divide both sides by -2, remember to reverse the inequality sign.

  • x <

Understanding Inequalities and Absolute Value Equations

An inequality with an absolute value will be in the form of |ax + b| > c, or | ax + b|< c. To solve |ax + b| > c, first drop the absolute value and create two separate inequalities with the word OR between them. To solve |ax + b|< c, first drop the absolute value and create two separate inequalities with the word AND between them. The first inequality will look just like the original inequality without the absolute value. For the second inequality, you must switch the inequality sign and change the sign of c. For example,

To solve |x + 3| > 5, first drop the absolute value sign and create two separate inequalities with the word OR between them.

  • x + 3 > 5 OR x + 3 < -5

Solve for x:

  • x > 2 OR x < -8

To solve |x + 3| < 5, first drop the absolute value sign and create two separate inequalities with the word AND between them:

  • x + 3 < 5 AND x + 3 > -5

Solve for x:

  • x < 2 AND x > -8

Understanding Functions

A function is a set of ordered pairs in which no two of the ordered pairs has the same x-value. In a function, each input (x-value) has exactly one output (y-value). An example of this relationship would be y = x2. Here, y is a function of x because for any value of x, there is exactly one value of y. However, x is not a function of y because for certain values of y, there is more than one value of x. The domain of a function refers to the x-values, whereas the range of a function refers to the y-values. If the values in the domain correspond to more than one value in the range, the relation is not a function.

For example: Let the function f be defined by f(x) = 2(3x). What is the value of f(5)?

Solve this problem by substituting 5 for x wherever x appears in the function:

  • f(x) = x2 – 3x

  • f(5) = (5) 2 – (3)(5)

  • f(5) = 25 – 15

  • f(5) = 10

Pearson IT Certification Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from Pearson IT Certification and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about Pearson IT Certification products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites; develop new products and services; conduct educational research; and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by Adobe Press. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.pearsonitcertification.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020