Q: Which of the following equations is a rational equation, and what is the main restriction when solving it?

$$\dfrac{x}{x-1} = 5$$; restriction: $$x \neq 1$$

Q: A chemist is mixing two solutions. The equation $$\dfrac{x+2}{x-1} = 5$$ models the concentration of a chemical. What is the solution, and what restriction must be checked?

$$x = 1.75$$; restriction: $$x \neq 1$$

Question 1

Which of the following equations is a rational equation, and what is the main restriction when solving it?

Accepted Answer

$$\dfrac{x}{x-1} = 5$$; restriction: $$x 
eq 1$$

Question 2

Solve the linear equation $$3(2 - 5x) = 4x + 25.$$

Accepted Answer

$$x = -1.1$$

Question 3

Which of the following best describes an inconsistent linear equation?

Accepted Answer

An equation that has no solution.

Question 4

Given the equation $$\dfrac{x}{4} + \dfrac{1}{6} = \dfrac{x}{3}$$, what is the solution?

Accepted Answer

$$x = 3$$

Question 5

Simplify $$\sqrt{-75}.$$

Accepted Answer

$$5i\sqrt{3}$$

Question 6

What is the standard form of the complex number $$\dfrac{9 + \sqrt{-16}}{3}$$?

Accepted Answer

$$3 + \dfrac{4}{3}i$$

Question 7

Find the product of $$(2 + 3i)(2 - 3i).$$

Accepted Answer

$$13$$

Question 8

Which of the following is the conjugate of $$-7 - i$$?

Accepted Answer

$$-7 + i$$

Question 9

Simplify $$i^{67}.$$

Accepted Answer

$$i$$

Question 10

A chemist is mixing two solutions. The equation $$\dfrac{x+2}{x-1} = 5$$ models the concentration of a chemical. What is the solution, and what restriction must be checked?

Accepted Answer

$$x = 1.75$$; restriction: $$x 
eq 1$$

Question 11

Express the sum $$5(4 + 7i) + 6(3 - 2i) in $$standard form.

Accepted Answer

$$38 + 17i$$

Question 12

Which of the following statements about complex conjugates is TRUE?

Accepted Answer

Multiplying a complex number by its conjugate always results in a real number.

Question 13

Divide and express in standard form: $$\dfrac{6 + i}{4 - 2i}.$$

Accepted Answer

$$\dfrac{1}{2} + i$$

Question 14

A medical researcher models the spread of a virus with the equation $$\dfrac{5}{x} = 4 + \dfrac{3}{x}$$, where $$x is $$the number of days. What is the solution for $$x$$?

Accepted Answer

$$x = 1$$

Question 15

Simplify $$i^{1003}.$$

Accepted Answer

$$i$$

Question 16

Which of the following is the standard form of the complex number $$3 + 2\sqrt{3}i$$?

Accepted Answer

$$3 + 2\sqrt{3}i$$

Question 17

A student solves $$x + 2 = 0$$ and gets $$x = -2. $$What type of solution is this?

Accepted Answer

Conditional

Question 18

Which of the following is the correct process to solve a linear equation with fractions?

Accepted Answer

Multiply both sides by the least common denominator, then solve as usual.

Question 19

If $$i = \sqrt{-1}$$, what is $$i^4$$?

Accepted Answer

$$1$$

Question 20

A physics equation is given by $$\dfrac{x-5}{x+4} = \dfrac{x-1}{x+4}. $$What is the solution set?

Accepted Answer

No solution

Precalculus Study Guide: Linear Equations and Complex Numbers

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