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~ Chapter 1 – Solving Equations and Inequalities ~

1.1 Linear Equations and ApplicationsEssential Question(s):

How do you solve linear equations? How do you solve linear word problems?

Vocabulary:

Algebraic EquationFormed by placing an equal sign between two algebraic expressionsCan be solved

Algebraic Expression Part of an algebraic equation. Can only be simplified.

Domain(replacement set)

The set of numbers that are permitted to replace the variable

Solution or Root Each element in the domain of the variable that makes the equation true

Solving an Equation Finding the complete solution set for the equation

Standard Form of a Linear Equation

Ax + By = C, where a ≠ 0

Transitions to College Math – Chapter 1 – Solving Equations and Inequalities Page #1

Properties of Equality: If a, b, and c are any real numbers and a = b, then

Addition Propertyof Equality Add the same value to BOTH sides of an equation

Subtraction Propertyof Equality Subtract the same value from BOTH sides of an equation

Multiplication Propertyof Equality Multiply the same value on BOTH sides of an equation

Division Propertyof Equality Divide by the same value BOTH sides of an equation

Substitution Property You may replace an expression by an equivalent expression without changing its value. “plugging into” an equation

Examples:

1. Solve.

2. Solve.

3. Solve for r.

4. Solve for s.


5. Find three consecutive odd integers such that 3 times their sum is 5 more than 8 times the middle one.

6. Find three consecutive odd integers such that the sum of the second, twice the first, and three times the third is 152.

6. The length of a rectangle is 3 ft less than 2 times its width. If the perimeter of the rectangle is 48 ft, find the dimensions of the rectangle.


w

2w –

1

P = 48 ft

7. How much pure antifreeze must be added to 12 gallons of 20% antifreeze to make a 40% antifreeze solution?

# of Gal % Antifreeze

Total Antifreeze

Orig. Sol. 12 20 240

Antifreeze x 100 100x

New Sol. 12 + x 40 40(12+x)

8. One computer printer can print a company's mailing labels in 40 minutes. A second printer would take 60 minutes to print the labels. How long would it take the two printers, operating together, to print the labels?

WR T WD

Printer 1 x

Printer 2 x


9. Bill's motorboat can travel 30 mi/h in still water. If the boat can travel 9 miles downstream in the same time it takes to travel 1 miles upstream, what is the rate of the river's current?

R T D

Upstream 30 – x t 9

Downstream 30 + x t 1

Downstream: Upstream:


1.2 Linear InequalitiesEssential Question(s):

How do you solve linear inequalities?

Vocabulary:

Trichotomy Property For any two real numbers a and b,

Interval The subset of real numbers that is the solution to an inequality

[ ] denotes…. A closed interval (endpoints included in the interval)

( ] denotes… A half-open interval (right endpoint included in the interval)

[ ) denotes… A half-open interval (left endpoint included in the interval)

( ) denotes… An open interval (left and right endpoints not included in the interval)

denotes… The union of sets A “OR” B. Combines all of set A with all of set B

denotes…The intersection of sets A “AND” B. Combines what is in common in sets A and B

Solution Set of an Inequality

The set of all values of the variable that make the inequality a true statement

Solving an inequality Finding the solution set of the inequality


Inequality Properties: If a, b, and c are any real numbers,

Transitive Property If a < b and b < c, then a < c

Addition Property If a < b, then a + c < b + c

Subtraction Property If a < b, then a – c < b – c

Multiplication PropertyIf a < b and c is positive, then ca < cb

If a < b and c is negative, then ca > cb

Division PropertyIf a < b and c is positive, then

If a < b and c is negative, then

Examples:1. Rewrite in inequality notation and graph on a

real number line.

(–4, 4]

–4 < x ≤ 4

2. Rewrite in interval notation and graph on a real number line.

1 ≤ x ≤ 5

[1, 5]


3. Write in interval notation and inequality notation.

(–1, ∞)

x > –1

4. Fill in the blanks with > or < to make the resulting statement true.

–4 < –2

and

–4 – 3 < –2 – 3

5. Graph and write as a single interval, if possible.

a. [–3, 6) [5, 8)

[5, 6]

b. [–3, 6) [5, 8)

[–3, 8)

6. For what real numbers x does the expression represent a real number?

7. Solve and graph.

3x – 7 ≥ x – 5

8. Solve and graph.


9. If F is the temperature in degrees Fahrenheit, then the temperature C in degrees Celsius is given

by the formula . For what Fahrenheit temperatures will the Celsius temperature be between –5 and 35, inclusive?


1.3 Absolute Value in Equations and InequalitiesEssential Question(s):

How do you solve absolute value equations? How do you solve absolute value inequalities?

Steps to Solve

1. Isolate the absolute value.

2. Write absolute value on left side.

3. Determine if “and” or “or.”

And :

Or :

4. Set up 2 equations.

Drop absolute value & solve.

Drop absolute value, flip the inequality, take the opposite and solve.

Examples:Equation/Inequality Inequality Notation Interval Notation Graph

Equality

Less Than

Greater Than

Notes:

If = negative number or <0 or negative number, there is no solution.

Translation: absolute values cannot be negative

If = 0 or , there is one solution.

If > negative number or , the solution is all real numbers.

Translation: absolute values are always positive


Solutions should ALWAYS be

graphed as well as

written in inequality AND

interval notation

-1 1

-1 1

-1 1

Examples:

Solve. How many solutions does each problem yield?

1.

2 sol’s

2.

1 sol

3. No solution

Solve. Solutions should be graphed as well as written in inequality and interval notations.

4. 5.

6. 7.


8. 9.

10. Solve. |x + 10| = 2x + 1

Case 1: Case 2:

So,


1.4 Complex NumbersEssential Question(s):

How do you add, subtract, multiply, and divide complex numbers?

Vocabulary: Anatomy of a Complex Number:

Standard Form of a Complex Number

Zero

Conjugate

Properties of Equality and Basic Operations:

EqualitySet reals=reals & imaginaries=imaginaries

AdditionAdd reals to reals & imaginaries to imaginaries

MultiplicationFOIL

Additive Identity

Additive Inverse

Multiplicative Identity 1

Principle Square Root of a Negative Number


Cyclic Properties of i:

i =

i 2 =

i 3 =

i 4 =

Real Part Imaginary Part

Examples:

1. For the complex number 4 – 2i, find…

a. the real part 4

b. the imaginary part -2i

c. the conjugate 4 + 2i

2. Add. Write the result in standard form.

(10 – 10i) + (1 + 7i)

11 – 3i

3. Subtract. Write the result in standard form.

(–4 – 8i) – (–1 – 7i)

4. Multiply. Write the result in standard form.

(5 – i)(4 + 3i)

11 – 3i

5. Evaluate. Write your answer in standard form. 6. Evaluate. Write your answer in standard form.


7. Divide. Write your answer in standard form. 8. Divide and write your answer in standard form.

9. Solve for x and y.


1.5 Quadratic Equations and ApplicationsEssential Question(s):

What are the ways to solve quadratic equations? How do you solve quadratic word problems?

Vocabulary:

Standard Form of a Quadratic Equation

Real root a real number solution of an equation

Imaginary root an imaginary number solution of an equation

Methods for solving quadratic equations:

I. Using Factoring and the Zero Property

Zero Property If m and n are complex numbers,

Examples: Solve by factoring.

1. 2.

II. Using the Square Root Property

Square Root Property If then

Examples: Solve using the square root property.

3. 4.


III. Completing the Square:

Steps to Solve

1. Transform the equation so that the constant term c is alone on the right side

2. If a is not equal to 1, then divide both sides by a.

3. Add to BOTH sides.

4. Factor the left side.

5. Use the square root property

6. Simplify.

Examples: Solve by completing the square.

5. 6.


IV. Using the Quadratic Formula:

Quadratic Formula

Discriminant

The DiscriminantValue of the Discriminant: Number of Roots: Graphical Representation:

Discriminant is positive.

Two Real Roots

If the discriminant is a perfect squarethe roots are rational

If the discriminant is not a perfect squarethe roots are irrational

Discriminant is zero.One Real Root

Discriminant is negative.Two Imaginary Conjugate Roots

Examples: Solve using the quadratic formula.

7. 8.


Find the number of real roots of each quadratic equation.

9. 10. 11.


32x

3x

3

312x

1x

Applications Vocabulary:

Demand the quantity of a product that people are willing to buy during some period of time

Price-Demand equation where q is the demand, a and b are constants, and p is the price

Revenue the amount of money received from the sale of q items at $p per item

Application Examples:

12. The product of two consecutive positive even integers is 168. Find the integers.

13. One positive number is 4 less than twice another positive number and their product is 96. Set-up an algebraic equation and solve it to find the two numbers.

14. Two technicians can complete a mailing in 3 hours when working together. Alone, one can complete the mailing 2 hours faster than the other. How long will it take each person to complete the mailing alone? Compute the answers to two decimal places.


Tech 1

Tech 2

15. A speedboat takes 3 hours longer to go 60 miles up a river than to return. If the boat cruises at 15 miles per hour in still water, what is the rate of the current?

R T D

Upstream 15 – x t + 3 60

Downstream 15 + x t 60

Upstream: Downstream:

16. The daily price-demand equation for whole milk in a chain of supermarkets is where p is the price per gallon and q is the number of gallons sold per day. Find the price(s) that will produce a revenue of $9,300.



1.6 Special Equation Solving TechniquesEssential Question(s):

How do you solve equations involving radicals and absolute value? How do you solve equations of the quadratic type?

I. Equations involving Radicals

When squaring both sides of an equation, extraneous solutions are introduced. You must check all possible solutions in the original equation to eliminate the extraneous solutions.

Examples:

1. Solve. 2. Solve.


II. Equations involving Absolute Value

Squaring both sides eliminates the need to consider both cases. However, you must still check all possible solutions in the original equation to eliminate the extraneous solutions.

Examples:

3. Solve. 4. Solve.

III. Equations involving the Quadratic Form

Use u-substitution.

Examples:

5. Solve. 6. Solve.


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