# Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job

# Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job

1 Conception

You do not need to memorize these conceptions, but make sure you understand them
and be

able to solve the problems contain them.

(1) Interest Problem:

Interest

Rate of interest

Simple Interest

Principal

(2) Mixture Problem: note the units of two materials

(3) Uniform Motion Problems:

Uniform Motion

Velocity

Distance

(4) Constant Rate Job Problems

Constant Rate Job

2 Formulas

(1) Simple Interest Formula (p. A63)

(2) Uniform Motion Formula (p. A65)

3 Solve Interest Problem

4 Solve Mixture Problem

5 Solve Uniform Motion Problems

6 Solve Constant Rate Job Problems

**A.9 Interval Notation; Solving Inequalities**

1 Use Interval Notation

Memorize Table 5 (p. A73), and test yourself by finishing the following form

Interval Type |
Interval Notation |
Inequality | Sketch the graph |

a<x<b | |||

a≤x≤b | |||

Half-open | [a, b) | a≤x<b | |

a<x≤b | |||

x≥a | |||

x>a | |||

x≤a | |||

x<a | |||

All Real Numbers |

2 Use Properties of Inequalities

Memorize the following four properties, and fill in the blanks with proper
inequality symbols .

(1) Nonnegative Property (p. A73)

For any real number a , a^{2} □ 0.

(2) Addition Property of Inequalities (p. A74)

For real numbers a, b, and c,

If a<b, then a+c □ b+c.

If a>b, then a+c □ b+c.

(3) Multiplication Properties for Inequalities (p. A74)

For real numbers a, b, and c,

If a<b and if c>0, then ac □ bc.

If a<b and if c>0, then ac □ bc.

If a>b and if c>0, then ac □ bc.

If a>b and if c<0, then ac □ bc.

(4) Reciprocal Property for Inequalities

If a>0, then 1/a □ 0. If
1/a> 0, then a □ 0.

If a<0, then 1/a □ 0. If
1/a < 0, then a □ 0.

3 Solve Inequalities

(1) Be familiar with the type of procedures leave the inequality symbol
unchanged or reversed.

Fill in the blanks with the proper inequality symbols (<,
>, ≤, ≥). (p. 75)

For expressions A, B, C, and D,

If ( A + C ) + B > D, then A + C + B □ D.

If A > B, then A + C □ B + C.

If A > B, then A – C □ B – C.

If A > B and C > 0, then .

If A > B and C≥0, then A× C □ B× C.

If A > B , then B □ A.

If A > B and C < 0, then .

If A > B and C≤0, then A× C □ B× C.

(2) Solve an Inequality

4 Solve Combined Inequalities

5 Solve Inequalities Involving Absolute Value

**A.10 nth Roots ; Rational Exponents**

1 n-th root

(1) Definition

Radical

Index

Radicand

Square root

Cube root

(2) Formula for (p. A82)

, if n≥3 is odd.

, if n≥2 is even.

(3) Using a calculator to approximate n-th root (p. A82)

(4) Properties of Radicals (p. A83)

(5) Simplify the radicals

a) Write the coefficient of the radicand as m-th (m≥n) power, i.e.,

b)

c) Divide m and k by n, we have

with integers and

d)

If

If

e)

If

If

f) Using the formulas in (2) to simplify the
term

(6) Combing Like Radicals

Note first simplify then combing. You CANT combine the like radicals as the
following

** (ERROR!)**

2 Rationalize Denominators

(1) If the denominator is monomial

a) Simplify the denominator first, and then get a new radical
(m<n and
k<n)

b) Multiply by

(2) If the denominator is binomial, A±B

Multiply by

3 Solve Radical Equations

(1) Move the radical to the left side, and the other terms to the right side

(2) Raise each side to the n-th power and solve

4 Simplify expressions with rational exponents

(1) Definition

(2) Writing expressions containing Fractional Exponents as Radicals

(3) Simplifying Expressions Containing Rational Exponents

Similar to the polynomial operations

(4) Writing an expression as a single quotient

(5) Factoring an Expression Containing Rational Exponents

**A.7 Complex Numbers ; Quadratic Equations
in the Complex Number System**

1 Definition (p. A54)

Imaginary unit | i | |

Complex number |
a+bi | |

Standard form | ||

Pure Imaginary number |
bi | |

Conjugate | ||

Principal square root |

2 Formulas (p. A55)

Equality of Complex Number |

a + bi = c + di if and only if a = c and b = d |

Sum of Complex Numbers |

Difference of Complex Numbers |

Product of Complex Numbers |

Powers of i (p. A58) |

3 Theorems (p. A56-A57)

The product of a complex number and its conjugate is a nonnegative real number. |

The Conjugate of a real number is the real number itself. |

The conjugate of the conjugate of a complex number is the number itself. |

The conjugate of the sum of two complex numbers equals the sum of their conjugates. |

The conjugate of the product of two complex numbers equals the product of their conjugates. |

Quadratic Formula: In the complex number system,
the solution of the quadratic equation
Where a, b, and c are real numbers and a≠0, are given by the formula |

x= |

4 Add, Subtract , Multiply, and Divide Complex Numbers

(1) Take the imaginary unit i as the square root
or a variable , and then
follow the rules of

the operations of the polynomials

(2) Substitute the powers of i with the above formulas (p. A58)

(3) Write in Standard Form

5 Solve quadratic equations in the complex number system

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