Nature of Mathematics - 12th Edition

Problems 1-2

Being able to adequately answer both of these questions is essential to being able to solve equations. Look at the essential ideas for the answers.

Problems 3-5

Problems 3a and 3b: Ask yourself what is being subtracted from x? Then add that number to both sides, as shown in Example 1.

Problems 3c and 3d: Ask yourself what is being added to x? Then, the opposite is to subtract that number from both sides.

Problems 4a and 4b: Ask yourself what is being divided into x? Then multiply both sides by that number, as shown in Example 1.

Problems 4c and 4d: Ask yourself what is x being multiplied by? Then, the opposite is to divide both sides by that number.

Problems 5a and 5b: Ask yourself what is x being multiplied by? Then, the opposite is to divide both sides by that number.

Problem 5c: Ask yourself what is x being multiplied by? Then, the opposite is to divide both sides by that number.

Problems 5d: What do you need to do to both sides? (see Example 1.)

Problems 6-9

Let x be the repeating decimal, and determine the number of digits that repeat. If it is one, multiply both sides by 10; if two digits repeat, multiply both sides by 100; if three digits repeat, multiply both sides by 1,000; and so on. Then subtract the original equation from both sides as shown

Problems 10-13

Look at the equations, first multiply the variable by a number, then add or subtract. To reverse these steps, first add or subtract, and then divide to reverse the multiplication. See Example 2d.

Problem 14: You can multiply first (using the distributive property), as shown in Example 2e, or you can divide first. That is, in part a, divide both sides by 4 and in part b divide both sides by 5.

Problems 15-16: What do you need to do to both sides? (see Example 1.) There are two thinks to do in Problem 13b: first multiply both sides by 3 and then divide both sides by 2.

Problem 14

You can multiply first (using the distributive property), as shown in Example 2e, or you can divide first. That is, in part a, divide both sides by 4 and in part b divide both sides by 5.

Problems 15-16

Look at the equations, first multiply the variable by a number, then add or subtract. To reverse these steps, first add or subtract, and then divide to reverse the multiplication. See Example 2d.

Problems 15-16: What do you need to do to both sides? (see Example 1.) There are two thinks to do in Problem 13b: first multiply both sides by 3 and then divide both sides by 2.

Problems 17-25

These problems require more than one operation, as shown in Example 2. First, make sure that the expression on the left side of the equal sign is simplified, and then make sure the expression on the right side of the equal sign is simplified.

Remember the simplified order of operations is first mult/divide and then add/sub.... to reverse this process (which is what we are doing when solving an equation), generally, you should add/sub the same number to both sides and then multiply/divide both sides by the same (nonzero) number.

Problems 26

Use the answers for Problems 10 to 25 to fill in the puzzle. For example, from Problem 8a, you know A = 5, so find any blanks in the puzzle which show the number 5 (there is only one); fill in the letter "A". (Yes, this means you need to work all of the problems.) If you have done these problems correctly, and if you have filled in the puzzle correctly, you will be able to read the secret message.

Problems 27-48

First, get a zero on one side.
Next, factor (if possible).
If it factors, use the zero-product rule (as shown in Example 4).
If it does not factor, use the quadratic formula (as shown in Example 5).

Problems 49-54

See Example 6 and the calculator comment.