Algebra Review for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide

Solve each equation:

1. x + 5 = 12
2. 3x + 6 = 18
3. 

Positive and Negative Numbers

Positive and negative numbers, also known as signed numbers, are best shown as points along the number line:

Numbers to the left of 0 are negative and those to the right are positive. Zero is neither negative nor positive. If a number is written without a sign, it is assumed to be positive. Notice that when you are on the negative side of the number line, numbers with bigger values are actually smaller. For example, –5 is less than –2. You come into contact with negative numbers more often than you might think; for example, very cold temperatures are recorded as negative numbers.

As you move to the right along the number line, the numbers get larger. Mathematically, to indicate that one number, say 4, is greater than another number, say –2, the greater than sign (>) is used:

4 > –2

On the other hand, to say that –2 is less than 4, we use the less than sign, (<):

–2 < 4

Arithmetic with Positive and Negative Numbers

The table below illustrates the rules for doing arithmetic with signed numbers. Notice that when a negative number follows an operation (as it does in the second example below), it is enclosed in parentheses to avoid confusion.

When more than one arithmetic operation appears, you must know the correct sequence in which to perform the operations. For example, do you know what to do first to calculate 2 + 3 × 4? You're right if you said, "multiply first." The correct answer is 14. If you add first, you will get the wrong answer of 20. The correct sequence of operations is:

Even when signed numbers appear in an equation, the step-by-step solution works exactly as it does for positive numbers. You just have to remember the arithmetic rules for negative numbers. For example, let's solve 14x + 2 = 5.

1. Subtract 2 from both sides:
2. Divide both sides by –14: –14x ÷ –14 = –7 ÷ –14

                                                       

Now try these problems with signed numbers.

4. 1 – 3 × (–4) = x
5. –3x + 6 = –18
6. + 3 = –7

Algebraic Expressions

An algebraic expression is a group of numbers, unknowns, and arithmetic operations, like: 3x – 2y. This one may be translated as, "3 times some number minus 2 times another number." To evaluate an algebraic expression, replace each variable with its value. For example, if x = 5 and y = 4, we would evaluate 3x – 2y as follows:

3(5) – 2(4) = 15 – 8 = 7

Evaluate these expressions.

7. 6a + 2b; a = 2 and b = –1
8. 3mn – 4m + 2n; m = 3 and n = –3
9. –2x – y + 4z; x = 5, y = –4, and z = 6
10. The volume of a cylinder is given by the formula V=πr²h, where r is the radius of the base and h is the height of the cylinder. What is the volume of a cylinder with a base radius of 3 and height of 4? (Leave π in your answer.)
11. If x = 3, what is the value of 3x – x?

Answers

1. 7
2. 4
3. 28
4. 13
5. 8
6. 40
7. 10
8. –45
9. 16
10. 36π
11. 6