**Question: Are pages 51-55 missing answers?**

Answer: No, they are not. Pages 51-55 are the introduction of Chapter 3, which includes instructions and examples. There are no exercises on those pages.

**Question: Should the answer to page 56, problem 9 be squareroot(7/5)?**

Answer: Actually, that's the same as the answer listed if you rationalize the denominator.

**Question: Could you please check the answer to page 69, exercise 7?**Answer: Even better, I will work out the solution for you so that you can follow along.

- 2x^2 - 6 = -8x^2 + 5x^2
- Add 3x^2 to both sides to get 5x^2 -6 = 0.
- Add 6 to both sides to get 5x^2 = 6.
- Divide both sides by 5 to get x^2 = 6/5.
- Square root both sides to get x = +/- sqrt(6/5).
- To put the answer in standard form, multiply the numerator and denominator both by the square root of 5 to get x = +/- sqrt(30) / 5.
- Note that sqrt(6/5) = sqrt(6) / sqrt(5), that sqrt(6*5) = sqrt(6) * sqrt(5), and that sqrt(5) * sqrt(5) = 5.
- The answer listed in the back of the book is +/- sqrt(30) / 5. If you use a calculator, this works out to 1.095445115, which is the same as sqrt(6/5).
- Plug 1.095445115 into the original equation to see that it works: 2*(1.095445115)^2 - 6 = -8*(1.095445115)^2 + 5*(1.095445115)^2. This simplifies to 2.4 - 6 = -9.6 + 6, which reduces to -3.6 = -3.6. The answer checks out.

**Question: Could you please check the answer to page 32, exercise 5?**Answer: Even better, I will work out the solution for you so that you can follow along.

- -x/5 - 3x/2 = -x/5 + 5/3
- Add x/5 to both sides to separate the variable terms (with x) from the constant term (5/3).
- -3x/2 = 5/3
- Multiply both sides by 2.
- -3x = 10/3
- Divide both sides by -3
- x = -10/9
- The answer listed in the back of the book is -10/9. You can also verify this answer with an online algebra solver, such as Mathematica offers.

**Question: Can you show me how to solve exercise 10 on page 39?**Answer: The section with fractions seems to draw many questions. All the more reason students need to practice this.

- 1 - 5x/2 = 3 - 2x
- Add 5x/2 to both sides to get 1 = 3 - 2x + 5x/2
- Write 2x as 4x/2 so that you can combine the variable terms with a common denominator: 1 = 3 - 4x/2 + 5x/2
- This simplifies to 1 = 3 + x/2.
- Isolate the unknown term by subtracting 3 from both sides: -2 = x/2.
- Multiply both sides by 2 to get -4 = x.

**TIPS ON CHECKING YOUR ANSWERS:** There can be multiple ways of expressing the same answer. Here are a few examples:

- Student's answer = 12/8. Listed answer = 3/2. Both are the same. It's standard form to reduce fractions when possible, so the book lists 3/2 as the answer. The student's answer of 12/8 is not technically incorrect.
- Student's answer = 1/sqrt(2). Listed answer = sqrt(2)/2. Both are the same. It's standard form to rationalize the denominator, so the book lists sqrt(2)/2 as the answer. Multiply both numerator and denominator by squareroot(2) to go from 1/sqrt(2) to sqrt(2)/2.
- Student's answer = sqrt(12). Listed answer = 2sqrt(3). Both are the same. It's standard form to factor out perfect squares, so the book lists 2sqrt(3) as the answer. Squareroot(12) = squareroot(3 x 4) = squareroot(3) x squareroot(4) = 2 x squareroot(3).

Students who don't reduce their fractions, factor out perfect squares, or rationalize their denominators may feel that the book's answer is incorrect, but that's not the case. Even some popular online equation solvers fail to rationalize the denominator, so a student who obtains 1/sqrt(2) as the answer and gets the same answer with an online equation solver might wonder if the book's answer of sqrt(2)/2 is incorrect. However, the book's answer is correct, and is also in standard form.

There is a simple way to check your answers. Enter both expressions in your calculator and display the answer as a decimal. This way, you can see that both 1/sqrt(2) and sqrt(2)/2 equal 0.70710678... If you reduce your fractions, factor out perfect squares, and rationalize your denominators, this won't be an issue.