Chapter 1

Preparing for Pre-Calculus

IN THIS CHAPTER

Brushing up on the order of operations

Solving equalities

Graphing equalities and inequalities

Finding distance, midpoint, and slope

Pre-calculus is the stepping stone for calculus. It's the final stepping stone after all those years of math: algebra I, geometry, algebra II, and trigonometry. Now all you need is pre-calculus to get to that ultimate goal - calculus. And as you may recall from your algebra II class, you were subjected to much of the same material you saw in algebra and even pre-algebra (just a couple steps up in terms of complexity - but really the same stuff). Pre-calculus begins with certain concepts that you need to be successful in any mathematics course.

If you feel you're already an expert at everything algebra, feel free to skip past this chapter and get the full swing of pre-calculus going. If you have any doubts or concerns, however, you may want to review; read on.

If you don't remember some of the concepts discussed in this chapter, or even in this book, you can pick up another For Dummies math book for review. The fundamentals are important. That's why they're called fundamentals. Take the time now to review and save yourself frustration and possible math errors in the future!

Reviewing Order of Operations: The Fun in Fundamentals

You can't put on your sock after you put on your shoe, can you? At least, you shouldn't! The same concept applies to mathematical operations. There's a specific order as to which operation you perform first, second, third, and so on. At this point, it should be second nature, but because the concept is so important (especially when you start doing more complex calculations), a quick review is worth it, starting with everyone's favorite mnemonic device.

Please excuse who? Oh, yeah, you remember this one - my dear Aunt Sally! The old mnemonic still stands, even as you get into more complicated problems. Please Excuse My Dear Aunt Sally is a mnemonic for the acronym PEMDAS, which stands for

• Parentheses (including absolute value, brackets, fraction lines, and radicals)
• Exponents (and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

The order in which you solve algebraic problems is very important. Always compute what's in the parentheses first, then move on to the exponents, followed by the multiplication and division (from left to right), and finally, the addition and subtraction (from left to right).

You should also have a good grasp on the properties of equality. If you do, you'll have an easier time simplifying expressions. Here are the properties:

• Reflexive property: . For example, .
• Symmetric property: If , then . For example, if , then .
• Transitive property: If and , then . For example, if and , then .
• Commutative property of addition: . For example, .
• Commutative property of multiplication: . For example, .
• Associative property of addition: . For example, .
• Associative property of multiplication: . For example, .
• Additive identity: . For example, .
• Multiplicative identity: . For example, .
• Additive inverse property: . For example, .
• Multiplicative inverse property: , as long as . For example, .
• Distributive property: . For example, .
• Multiplicative property of zero: . For example, .
• Zero product property: If , then or . For example, if , then or .

Following are a couple examples so you can see the order of operations and the properties of equality in action before diving into some practice questions.

Q. Simplify:

A. The answer is 5.

Following the order of operations, simplify everything in parentheses first. (Remember that radicals and absolute value bars act like parentheses, so do operations within them first before simplifying the radicals or taking the absolute value.)

Simplify the parentheses by taking the square root of 25 and subtracting the result from 3; find the absolute value of :

Now you can deal with the exponents by squaring the 6 and the :

Although they're not written, parentheses are implied around the terms above and below a fraction bar. Therefore, you must simplify the numerator and denominator before dividing the terms following the order of operations:

Q. Simplify:

A. The answer is 3.

Using the associative property and the commutative property of addition, rewrite the expression to make the fractions easier to add. Then add the fractions with the common denominators.

Then reduce the resulting fraction and change the fractions in the numerator and denominator to equivalent fractions with common denominators:

Adding the fractions, you get:

To simplify the complex fraction, you multiply the numerator by the reciprocal of the denominator:

1 Simplify:

2 Simplify:

3 Simplify:

4 Simplify:

Keeping Your Balance While Solving Equalities

Just as simplifying expressions is a basic process in pre-algebra, solving for variables is the basis of algebra. And both are essential to the more complex concepts covered in pre-calculus.

Solving linear equations with the general format of , where a, b, and c are constants, is relatively easy using the properties of numbers. The goal, of course, is to isolate the variable, x.

One type of equation you can't forget is the absolute value equation. The absolute value of a number is defined as its distance from 0. In other words, . This definition is a piecewise function with two rules: one where the quantity inside the absolute value bars is positive and another where it's negative. To solve these equations, you must isolate the absolute value term and then set the quantity inside the absolute value bars to the positive and negative values (see the second example question that follows).

Check out the following examples or skip ahead to the practice questions if you think you're ready to tackle them.

Q. Solve for x:

A.

First, using the distributive property, distribute the 3 and the to get . Then combine like terms and solve using algebra, like so: giving you , and, finally, .

Q. Solve for x:

A.

Isolate the absolute value by adding 16 to each side, giving you . One solution comes when you assume that the quantity inside the absolute value bars is positive: . This gives you the answer . The second solution comes from assuming that the quantity inside the absolute value bars is negative: . This becomes , then , and finally .

5 Solve:

6 Solve:

7 Solve:

8 Solve:

9 Solve:

10 Solve:

When Your Image Really Counts: Graphing Equalities and Inequalities

Graphs are visual representations of mathematical equations. In pre-calculus, you'll be introduced to many new mathematical equations and then be expected to graph them. You will have plenty of practice graphing these equations when you read the material involving the more complex equations. In the meantime, it's important to practice the basics: graphing linear equations and inequalities.

The graphs of linear equations and inequalities exist on the Cartesian coordinate system, which is made up of two axes: the horizontal, or x-axis, and the vertical, or y-axis. Each point on the coordinate plane is called a Cartesian coordinate pair and has an x coordinate and a y coordinate. The notation for any point on the coordinate plane looks like this: (x, y). A set of these ordered pairs that can be graphed on a coordinate plane is called a relation. The x values of a relation are its domain, and the y values are its range. For example, the domain of the relation is , and the range is .

You can graph a linear equation using two points or by using the slope-intercept form. The same can be used when graphing linear inequalities. These approaches are reviewed in the following sections.

Graphing with two points

To graph a line using two points, choose two numbers and plug them into the equation to solve for the range (y) values. After you plot these points (x, y) on the coordinate plane, you can draw the line through the points.

A nice alternative is to use the two intercepts, the points that fall on the x- or y-axes. To find the x-intercept (x, 0), plug in 0 for y and solve for x. To find the...