CliffsNotes CBEST - BTPS TESTING Ph.D., Jerry Bobrow Ph.D. & 8 more 2021

Algebra review
Algebra review

As you approach algebra questions on the CBEST, look for patterns and relationships among the numbers, variables, and their operations. You may not be aware of it, but you already use algebraic skills every day. Any time you analyze unknown quantities, you are using a part of your brain that accesses the reasoning skills necessary to perform algebraic equations.

Let’s take a look at an everyday scenario to illustrate how the relationships among numbers, variables, and their operations help you solve unknown variables. For example, suppose you need to attend a school district conference and want to estimate how long it will take (the unknown variable is time = t) to get to the conference center. The conference center is 30 miles from your home (distance = d), and you travel at a steady rate of 60 mph (rate of speed = r). You could have easily calculated the answer in your head, but the algebraic expression known as the time-distance formula, d = r · t (or t = d ÷ r or r = d ÷ t) will always help you solve this type of problem. At a steady rate of speed, it will take 30 minutes to travel to the university. This is algebra!

Seeing the Structure in Expressions

Analyzing algebraic equations depends on your knowledge of arithmetic operations. Your goal should be to have a solid understanding of the arithmetic topics covered in Chapter 4 to prepare you for the algebra topics.

Variables

Think of algebra as a language that uses letters (variables) such as x, y, a, b, m, n, and so on to represent unknown rational numbers. A variable, therefore, is a letter used to represent a number until the numeric value is known. The importance of using variables in real-life scenarios is endless. For example, when ordering student textbooks for your classroom of 25 students, you may need to order an unknown number of additional textbooks (variable x) in the event additional students register the first week of school, or 25 + x.

Variables are often used to change verbal expressions into algebraic expressions. To solve problems using algebraic expressions, you will need to analyze the structural relationship between variables (unknown quantities) and numbers (known quantities).

In the algebraic expression x + 4 = 19, some unknown number (represented by x) plus 4 equals 19. Using basic arithmetic, it is easy to see that 15 + 4 = 19.

Equations

An equation is a relationship between numbers and/or symbols. It helps to remember that an equation is like a balance scale, with the equal sign (=) serving as the fulcrum, or center. Thus, if you do the same thing on both sides of the equal sign (say, add 5 to each side), the equation stays balanced.

For example, to solve the equation x — 5 = 23, you must get x by itself on one side; therefore, add 5 to both sides:

In the same manner, you may subtract, multiply, or divide both sides of an equation by the same (nonzero) number, and the equation will not change. Sometimes you may have to use more than one step to solve for an unknown.

Examples:

1. Solve for x: 3x + 4 = 19

Subtract 4 from both sides to get 3x by itself on one side:

Then divide both sides by 3 to get x:

TEST TIP: Remember, when solving an equation, the goal is to get the variable x on one side of the equation by itself. Use inverse (opposite) operations to “undo” each other to get the variable x alone. (This applies to addition, subtraction, multiplication, and division.)

2. If 3x + 1 = 16, what is the value of x — 4?

A. 19

B. 16

C. 5

D. 1

E. —1

To help you focus on the question, you may have jotted down “x — 4.” Note that solving the original equation only tells you the value of x.

Notice that choice C is 5, but this is incorrect because the question asks, “What is the value of x — 4,” not just x. To continue the problem, replace x with 5 and solve.

The correct answer is choice D.

Algebraic Multiplication

Algebraic multiplication refers to the common multiplication symbols that appear in algebraic equations.

Multiplication is implied when two or more variables (or variables and numbers) are placed next to each other. In the first example below, the letter x is the variable, and 5 (the coefficient) is the number used to multiply the variable.

5x (reads 5 times x)

ab (reads a times b)

4ab (reads 4 times a times b)

Perform multiplication when there are parentheses around one variable (or number).

(a)b or (a)4

a(b) or a(4)

(a)(b) or (a)(4)

Perform multiplication when there is a raised dot between variables and/or numbers.

a · b

5 · 4

Evaluating Equations

To evaluate an equation, evaluate, replace, or substitute the numerical value in the variable and simplify the expression using order of operations.

Example:

Evaluate: 2x² + 3y + 6 if x = 2 and y = 9.

Monomials and Polynomials

A monomial is an algebraic expression that consists of only one term (with no addition or subtraction signs). For example, 9x, 4a², and 3xz² are all monomials.

A polynomial is an algebraic expression made up of one or more terms. Terms are separated by addition or subtraction signs (no equal signs). For example, x + y, y² — x², and 15x² + 3x + 5y² are all polynomials.

To add or subtract monomials, follow the same rules as with regular signed numbers, provided that the terms are alike.

Examples:

1. 3x + 2x = 5x

2. 

Reasoning with Algebra

Algebraic Word Problems

This section covers different real-world relationships in the form of algebraic word problems.

Translating Word Expressions into Algebraic Expressions

To solve real-life algebraic word problems, you must be able to convert English words that describe a situation or scenario into simple algebraic equations. As discussed in the Arithmetic chapter on pp. 88—89, here are some verbal expressions that signal a math operation.

· Addition: sum, plus, more than, greater than, increase, rise

· Subtraction: minus, difference, less than, decrease, reduce

· Multiplication: times, multiplied by, of, product, twice

· Division: divided by, ratio, half, quotient

Examples:

Solving Algebraic Word Problems

The most common forms of algebraic word problems are distance and time, work, mixture, motion, and age-related scenarios. The key to solving these types of problems is to identify specific details about what the question is asking. A common mistake the hurried test-taker makes is to quickly read the question and rush to a solution. Algebraic word problems can be misleading unless you carefully organize the words into math variables and numbers.

Examples:

1. Juan can do a certain job alone in 8 hours, while it would take Susan 12 hours to do the same job alone. If they work together, how long should it take to do the job?

A. 20 hours

B. 10 hours

C. At least 5 hours but less than 10 hours

D. Less than 5 hours

E. More than 5 hours

Use reasoning and common sense to eliminate choices A and B, since together these choices, 20 hours and 10 hours, will require more time than the person who already completes the job in the least amount of time.

Using a fast method.

Using algebra. Let x be the amount of time it takes together.

If Juan needs 8 hours to do the job, then in 1 hour he does of the job. If Susan needs 12 hours to do the job, then in 1 hour she does of the job. If together they need x hours to do the job, then if they have 1 hour together, they will be able to complete of the job.

Therefore, . Now solve this equation.

Together they will need less than 5 hours. The correct answer is choice D.

2. In 6 years, men like Antonio Stradivari were able to build 58 violins. Considering all violin-makers worked at the same speed, how many complete violins could be built in 11 years by one man?

The first step to answering this question is to write down what is known: how many complete violins (note the word complete). Now, using the information given in the problem, set up a proportion equation and solve for x.

The question asks for the number of complete violins that one man could build in 11 years. Therefore, you must round down to the nearest whole number. The correct answer is 106.